Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnaordex GIF version

Theorem nnaordex 6100
 Description: Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaordex ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnaordex
Dummy variables 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2101 . . . . . 6 (𝑏 = 𝐵 → (𝐴𝑏𝐴𝐵))
2 eqeq2 2049 . . . . . . . 8 (𝑏 = 𝐵 → ((𝐴 +𝑜 𝑥) = 𝑏 ↔ (𝐴 +𝑜 𝑥) = 𝐵))
32anbi2d 437 . . . . . . 7 (𝑏 = 𝐵 → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
43rexbidv 2327 . . . . . 6 (𝑏 = 𝐵 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
51, 4imbi12d 223 . . . . 5 (𝑏 = 𝐵 → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏)) ↔ (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
65imbi2d 219 . . . 4 (𝑏 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏))) ↔ (𝐴 ∈ ω → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))))
7 eleq2 2101 . . . . . 6 (𝑏 = ∅ → (𝐴𝑏𝐴 ∈ ∅))
8 eqeq2 2049 . . . . . . . 8 (𝑏 = ∅ → ((𝐴 +𝑜 𝑥) = 𝑏 ↔ (𝐴 +𝑜 𝑥) = ∅))
98anbi2d 437 . . . . . . 7 (𝑏 = ∅ → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅)))
109rexbidv 2327 . . . . . 6 (𝑏 = ∅ → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅)))
117, 10imbi12d 223 . . . . 5 (𝑏 = ∅ → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏)) ↔ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅))))
12 eleq2 2101 . . . . . 6 (𝑏 = 𝑦 → (𝐴𝑏𝐴𝑦))
13 eqeq2 2049 . . . . . . . 8 (𝑏 = 𝑦 → ((𝐴 +𝑜 𝑥) = 𝑏 ↔ (𝐴 +𝑜 𝑥) = 𝑦))
1413anbi2d 437 . . . . . . 7 (𝑏 = 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)))
1514rexbidv 2327 . . . . . 6 (𝑏 = 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)))
1612, 15imbi12d 223 . . . . 5 (𝑏 = 𝑦 → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏)) ↔ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))))
17 eleq2 2101 . . . . . 6 (𝑏 = suc 𝑦 → (𝐴𝑏𝐴 ∈ suc 𝑦))
18 eqeq2 2049 . . . . . . . 8 (𝑏 = suc 𝑦 → ((𝐴 +𝑜 𝑥) = 𝑏 ↔ (𝐴 +𝑜 𝑥) = suc 𝑦))
1918anbi2d 437 . . . . . . 7 (𝑏 = suc 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
2019rexbidv 2327 . . . . . 6 (𝑏 = suc 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
2117, 20imbi12d 223 . . . . 5 (𝑏 = suc 𝑦 → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏)) ↔ (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))))
22 noel 3228 . . . . . . 7 ¬ 𝐴 ∈ ∅
2322pm2.21i 575 . . . . . 6 (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅))
2423a1i 9 . . . . 5 (𝐴 ∈ ω → (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅)))
25 elsuci 4140 . . . . . . 7 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
26 simpr 103 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)))
27 peano2 4318 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
2827ad2antlr 458 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)) → suc 𝑥 ∈ ω)
29 elelsuc 4146 . . . . . . . . . . . . . . . . 17 (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥)
3029a1i 9 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥))
31 nnasuc 6055 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
32 suceq 4139 . . . . . . . . . . . . . . . . . 18 ((𝐴 +𝑜 𝑥) = 𝑦 → suc (𝐴 +𝑜 𝑥) = suc 𝑦)
3331, 32sylan9eq 2092 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝐴 +𝑜 suc 𝑥) = suc 𝑦)
3433ex 108 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝑦 → (𝐴 +𝑜 suc 𝑥) = suc 𝑦))
3530, 34anim12d 318 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (∅ ∈ suc 𝑥 ∧ (𝐴 +𝑜 suc 𝑥) = suc 𝑦)))
3635imp 115 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)) → (∅ ∈ suc 𝑥 ∧ (𝐴 +𝑜 suc 𝑥) = suc 𝑦))
37 eleq2 2101 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ suc 𝑥))
38 oveq2 5520 . . . . . . . . . . . . . . . . 17 (𝑧 = suc 𝑥 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 suc 𝑥))
3938eqeq1d 2048 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → ((𝐴 +𝑜 𝑧) = suc 𝑦 ↔ (𝐴 +𝑜 suc 𝑥) = suc 𝑦))
4037, 39anbi12d 442 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +𝑜 suc 𝑥) = suc 𝑦)))
4140rspcev 2656 . . . . . . . . . . . . . 14 ((suc 𝑥 ∈ ω ∧ (∅ ∈ suc 𝑥 ∧ (𝐴 +𝑜 suc 𝑥) = suc 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦))
4228, 36, 41syl2anc 391 . . . . . . . . . . . . 13 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦))
4342ex 108 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦)))
4443rexlimdva 2433 . . . . . . . . . . 11 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦)))
45 eleq2 2101 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝑥))
46 oveq2 5520 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑥))
4746eqeq1d 2048 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝐴 +𝑜 𝑧) = suc 𝑦 ↔ (𝐴 +𝑜 𝑥) = suc 𝑦))
4845, 47anbi12d 442 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
4948cbvrexv 2534 . . . . . . . . . . 11 (∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))
5044, 49syl6ib 150 . . . . . . . . . 10 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
5150ad2antlr 458 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
5226, 51syld 40 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
53 0lt1o 6023 . . . . . . . . . . . 12 ∅ ∈ 1𝑜
5453a1i 9 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∅ ∈ 1𝑜)
55 nnon 4332 . . . . . . . . . . . . 13 (𝐴 ∈ ω → 𝐴 ∈ On)
56 oa1suc 6047 . . . . . . . . . . . . 13 (𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc 𝐴)
5755, 56syl 14 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +𝑜 1𝑜) = suc 𝐴)
58 suceq 4139 . . . . . . . . . . . 12 (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
5957, 58sylan9eq 2092 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → (𝐴 +𝑜 1𝑜) = suc 𝑦)
60 1onn 6093 . . . . . . . . . . . 12 1𝑜 ∈ ω
61 eleq2 2101 . . . . . . . . . . . . . 14 (𝑥 = 1𝑜 → (∅ ∈ 𝑥 ↔ ∅ ∈ 1𝑜))
62 oveq2 5520 . . . . . . . . . . . . . . 15 (𝑥 = 1𝑜 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 1𝑜))
6362eqeq1d 2048 . . . . . . . . . . . . . 14 (𝑥 = 1𝑜 → ((𝐴 +𝑜 𝑥) = suc 𝑦 ↔ (𝐴 +𝑜 1𝑜) = suc 𝑦))
6461, 63anbi12d 442 . . . . . . . . . . . . 13 (𝑥 = 1𝑜 → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦) ↔ (∅ ∈ 1𝑜 ∧ (𝐴 +𝑜 1𝑜) = suc 𝑦)))
6564rspcev 2656 . . . . . . . . . . . 12 ((1𝑜 ∈ ω ∧ (∅ ∈ 1𝑜 ∧ (𝐴 +𝑜 1𝑜) = suc 𝑦)) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))
6660, 65mpan 400 . . . . . . . . . . 11 ((∅ ∈ 1𝑜 ∧ (𝐴 +𝑜 1𝑜) = suc 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))
6754, 59, 66syl2anc 391 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))
6867ex 108 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
6968ad2antlr 458 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
7052, 69jaod 637 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → ((𝐴𝑦𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
7125, 70syl5 28 . . . . . 6 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
7271exp31 346 . . . . 5 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)) → (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))))
7311, 16, 21, 24, 72finds2 4324 . . . 4 (𝑏 ∈ ω → (𝐴 ∈ ω → (𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏))))
746, 73vtoclga 2619 . . 3 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
7574impcom 116 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
76 peano1 4317 . . . . . . . . 9 ∅ ∈ ω
77 nnaord 6082 . . . . . . . . 9 ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
7876, 77mp3an1 1219 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
7978ancoms 255 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
80 nna0 6053 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
8180adantr 261 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 ∅) = 𝐴)
8281eleq1d 2106 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴 ∈ (𝐴 +𝑜 𝑥)))
8379, 82bitrd 177 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +𝑜 𝑥)))
8483anbi1d 438 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) ↔ (𝐴 ∈ (𝐴 +𝑜 𝑥) ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
85 eleq2 2101 . . . . . 6 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
8685biimpac 282 . . . . 5 ((𝐴 ∈ (𝐴 +𝑜 𝑥) ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵)
8784, 86syl6bi 152 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
8887rexlimdva 2433 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
8988adantr 261 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
9075, 89impbid 120 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  ∅c0 3224  Oncon0 4100  suc csuc 4102  ωcom 4313  (class class class)co 5512  1𝑜c1o 5994   +𝑜 coa 5998 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005 This theorem is referenced by:  nnawordex  6101  ltexpi  6435
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