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Theorem nnawordex 6101
 Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnawordex ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnawordex
StepHypRef Expression
1 nntri3or 6072 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
213adant3 924 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3 nnaordex 6100 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
4 simpr 103 . . . . . . . 8 ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (𝐴 +𝑜 𝑥) = 𝐵)
54reximi 2416 . . . . . . 7 (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
63, 5syl6bi 152 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
763adant3 924 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
8 nna0 6053 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
983ad2ant1 925 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
10 eqeq2 2049 . . . . . . 7 (𝐴 = 𝐵 → ((𝐴 +𝑜 ∅) = 𝐴 ↔ (𝐴 +𝑜 ∅) = 𝐵))
119, 10syl5ibcom 144 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 = 𝐵 → (𝐴 +𝑜 ∅) = 𝐵))
12 peano1 4317 . . . . . . 7 ∅ ∈ ω
13 oveq2 5520 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
1413eqeq1d 2048 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 ∅) = 𝐵))
1514rspcev 2656 . . . . . . 7 ((∅ ∈ ω ∧ (𝐴 +𝑜 ∅) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
1612, 15mpan 400 . . . . . 6 ((𝐴 +𝑜 ∅) = 𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
1711, 16syl6 29 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 = 𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
18 nntri1 6074 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1918biimp3a 1235 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ¬ 𝐵𝐴)
2019pm2.21d 549 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵𝐴 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
217, 17, 203jaod 1199 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
222, 21mpd 13 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
23223expia 1106 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
24 nnaword1 6086 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
25 sseq2 2967 . . . . 5 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
2624, 25syl5ibcom 144 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
2726rexlimdva 2433 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
2827adantr 261 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
2923, 28impbid 120 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ w3o 884   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917  ∅c0 3224  ωcom 4313  (class class class)co 5512   +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:  prarloclemn  6597
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