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Theorem nnaordex 6011
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) → (A Bx 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 2083 . . . . . 6 (𝑏 = B → (A 𝑏A B))
2 eqeq2 2031 . . . . . . . 8 (𝑏 = B → ((A +𝑜 x) = 𝑏 ↔ (A +𝑜 x) = B))
32anbi2d 440 . . . . . . 7 (𝑏 = B → ((∅ x (A +𝑜 x) = 𝑏) ↔ (∅ x (A +𝑜 x) = B)))
43rexbidv 2305 . . . . . 6 (𝑏 = B → (x 𝜔 (∅ x (A +𝑜 x) = 𝑏) ↔ x 𝜔 (∅ x (A +𝑜 x) = B)))
51, 4imbi12d 223 . . . . 5 (𝑏 = B → ((A 𝑏x 𝜔 (∅ x (A +𝑜 x) = 𝑏)) ↔ (A Bx 𝜔 (∅ x (A +𝑜 x) = B))))
65imbi2d 219 . . . 4 (𝑏 = B → ((A 𝜔 → (A 𝑏x 𝜔 (∅ x (A +𝑜 x) = 𝑏))) ↔ (A 𝜔 → (A Bx 𝜔 (∅ x (A +𝑜 x) = B)))))
7 eleq2 2083 . . . . . 6 (𝑏 = ∅ → (A 𝑏A ∅))
8 eqeq2 2031 . . . . . . . 8 (𝑏 = ∅ → ((A +𝑜 x) = 𝑏 ↔ (A +𝑜 x) = ∅))
98anbi2d 440 . . . . . . 7 (𝑏 = ∅ → ((∅ x (A +𝑜 x) = 𝑏) ↔ (∅ x (A +𝑜 x) = ∅)))
109rexbidv 2305 . . . . . 6 (𝑏 = ∅ → (x 𝜔 (∅ x (A +𝑜 x) = 𝑏) ↔ x 𝜔 (∅ x (A +𝑜 x) = ∅)))
117, 10imbi12d 223 . . . . 5 (𝑏 = ∅ → ((A 𝑏x 𝜔 (∅ x (A +𝑜 x) = 𝑏)) ↔ (A ∅ → x 𝜔 (∅ x (A +𝑜 x) = ∅))))
12 eleq2 2083 . . . . . 6 (𝑏 = y → (A 𝑏A y))
13 eqeq2 2031 . . . . . . . 8 (𝑏 = y → ((A +𝑜 x) = 𝑏 ↔ (A +𝑜 x) = y))
1413anbi2d 440 . . . . . . 7 (𝑏 = y → ((∅ x (A +𝑜 x) = 𝑏) ↔ (∅ x (A +𝑜 x) = y)))
1514rexbidv 2305 . . . . . 6 (𝑏 = y → (x 𝜔 (∅ x (A +𝑜 x) = 𝑏) ↔ x 𝜔 (∅ x (A +𝑜 x) = y)))
1612, 15imbi12d 223 . . . . 5 (𝑏 = y → ((A 𝑏x 𝜔 (∅ x (A +𝑜 x) = 𝑏)) ↔ (A yx 𝜔 (∅ x (A +𝑜 x) = y))))
17 eleq2 2083 . . . . . 6 (𝑏 = suc y → (A 𝑏A suc y))
18 eqeq2 2031 . . . . . . . 8 (𝑏 = suc y → ((A +𝑜 x) = 𝑏 ↔ (A +𝑜 x) = suc y))
1918anbi2d 440 . . . . . . 7 (𝑏 = suc y → ((∅ x (A +𝑜 x) = 𝑏) ↔ (∅ x (A +𝑜 x) = suc y)))
2019rexbidv 2305 . . . . . 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 → ((A 𝑏x 𝜔 (∅ x (A +𝑜 x) = 𝑏)) ↔ (A suc yx 𝜔 (∅ x (A +𝑜 x) = suc y))))
22 noel 3205 . . . . . . 7 ¬ A
2322pm2.21i 562 . . . . . 6 (A ∅ → x 𝜔 (∅ x (A +𝑜 x) = ∅))
2423a1i 9 . . . . 5 (A 𝜔 → (A ∅ → x 𝜔 (∅ x (A +𝑜 x) = ∅)))
25 elsuci 4089 . . . . . . 7 (A suc y → (A y A = y))
26 ax-ia2 100 . . . . . . . . 9 (((y 𝜔 A 𝜔) (A yx 𝜔 (∅ x (A +𝑜 x) = y))) → (A yx 𝜔 (∅ x (A +𝑜 x) = y)))
27 peano2 4245 . . . . . . . . . . . . . . 15 (x 𝜔 → suc x 𝜔)
2827ad2antlr 462 . . . . . . . . . . . . . 14 (((A 𝜔 x 𝜔) (∅ x (A +𝑜 x) = y)) → suc x 𝜔)
29 elelsuc 4095 . . . . . . . . . . . . . . . . 17 (∅ x → ∅ suc x)
3029a1i 9 . . . . . . . . . . . . . . . 16 ((A 𝜔 x 𝜔) → (∅ x → ∅ suc x))
31 nnasuc 5970 . . . . . . . . . . . . . . . . . 18 ((A 𝜔 x 𝜔) → (A +𝑜 suc x) = suc (A +𝑜 x))
32 suceq 4088 . . . . . . . . . . . . . . . . . 18 ((A +𝑜 x) = y → suc (A +𝑜 x) = suc y)
3331, 32sylan9eq 2074 . . . . . . . . . . . . . . . . 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 2083 . . . . . . . . . . . . . . . 16 (z = suc x → (∅ z ↔ ∅ suc x))
38 oveq2 5444 . . . . . . . . . . . . . . . . 17 (z = suc x → (A +𝑜 z) = (A +𝑜 suc x))
3938eqeq1d 2030 . . . . . . . . . . . . . . . 16 (z = suc x → ((A +𝑜 z) = suc y ↔ (A +𝑜 suc x) = suc y))
4037, 39anbi12d 445 . . . . . . . . . . . . . . 15 (z = suc x → ((∅ z (A +𝑜 z) = suc y) ↔ (∅ suc x (A +𝑜 suc x) = suc y)))
4140rspcev 2633 . . . . . . . . . . . . . 14 ((suc x 𝜔 (∅ suc x (A +𝑜 suc x) = suc y)) → z 𝜔 (∅ z (A +𝑜 z) = suc y))
4228, 36, 41syl2anc 393 . . . . . . . . . . . . 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 2411 . . . . . . . . . . 11 (A 𝜔 → (x 𝜔 (∅ x (A +𝑜 x) = y) → z 𝜔 (∅ z (A +𝑜 z) = suc y)))
45 eleq2 2083 . . . . . . . . . . . . 13 (z = x → (∅ z ↔ ∅ x))
46 oveq2 5444 . . . . . . . . . . . . . 14 (z = x → (A +𝑜 z) = (A +𝑜 x))
4746eqeq1d 2030 . . . . . . . . . . . . 13 (z = x → ((A +𝑜 z) = suc y ↔ (A +𝑜 x) = suc y))
4845, 47anbi12d 445 . . . . . . . . . . . 12 (z = x → ((∅ z (A +𝑜 z) = suc y) ↔ (∅ x (A +𝑜 x) = suc y)))
4948cbvrexv 2512 . . . . . . . . . . 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 462 . . . . . . . . 9 (((y 𝜔 A 𝜔) (A yx 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) (A yx 𝜔 (∅ x (A +𝑜 x) = y))) → (A yx 𝜔 (∅ x (A +𝑜 x) = suc y)))
53 0lt1o 5938 . . . . . . . . . . . 12 1𝑜
5453a1i 9 . . . . . . . . . . 11 ((A 𝜔 A = y) → ∅ 1𝑜)
55 nnon 4259 . . . . . . . . . . . . 13 (A 𝜔 → A On)
56 oa1suc 5962 . . . . . . . . . . . . 13 (A On → (A +𝑜 1𝑜) = suc A)
5755, 56syl 14 . . . . . . . . . . . 12 (A 𝜔 → (A +𝑜 1𝑜) = suc A)
58 suceq 4088 . . . . . . . . . . . 12 (A = y → suc A = suc y)
5957, 58sylan9eq 2074 . . . . . . . . . . 11 ((A 𝜔 A = y) → (A +𝑜 1𝑜) = suc y)
60 1onn 6004 . . . . . . . . . . . 12 1𝑜 𝜔
61 eleq2 2083 . . . . . . . . . . . . . 14 (x = 1𝑜 → (∅ x ↔ ∅ 1𝑜))
62 oveq2 5444 . . . . . . . . . . . . . . 15 (x = 1𝑜 → (A +𝑜 x) = (A +𝑜 1𝑜))
6362eqeq1d 2030 . . . . . . . . . . . . . 14 (x = 1𝑜 → ((A +𝑜 x) = suc y ↔ (A +𝑜 1𝑜) = suc y))
6461, 63anbi12d 445 . . . . . . . . . . . . 13 (x = 1𝑜 → ((∅ x (A +𝑜 x) = suc y) ↔ (∅ 1𝑜 (A +𝑜 1𝑜) = suc y)))
6564rspcev 2633 . . . . . . . . . . . 12 ((1𝑜 𝜔 (∅ 1𝑜 (A +𝑜 1𝑜) = suc y)) → x 𝜔 (∅ x (A +𝑜 x) = suc y))
6660, 65mpan 402 . . . . . . . . . . 11 ((∅ 1𝑜 (A +𝑜 1𝑜) = suc y) → x 𝜔 (∅ x (A +𝑜 x) = suc y))
6754, 59, 66syl2anc 393 . . . . . . . . . 10 ((A 𝜔 A = y) → x 𝜔 (∅ x (A +𝑜 x) = suc y))
6867ex 108 . . . . . . . . 9 (A 𝜔 → (A = yx 𝜔 (∅ x (A +𝑜 x) = suc y)))
6968ad2antlr 462 . . . . . . . 8 (((y 𝜔 A 𝜔) (A yx 𝜔 (∅ x (A +𝑜 x) = y))) → (A = yx 𝜔 (∅ x (A +𝑜 x) = suc y)))
7052, 69jaod 624 . . . . . . 7 (((y 𝜔 A 𝜔) (A yx 𝜔 (∅ x (A +𝑜 x) = y))) → ((A y A = y) → x 𝜔 (∅ x (A +𝑜 x) = suc y)))
7125, 70syl5 28 . . . . . 6 (((y 𝜔 A 𝜔) (A yx 𝜔 (∅ x (A +𝑜 x) = y))) → (A suc yx 𝜔 (∅ x (A +𝑜 x) = suc y)))
7271exp31 346 . . . . 5 (y 𝜔 → (A 𝜔 → ((A yx 𝜔 (∅ x (A +𝑜 x) = y)) → (A suc yx 𝜔 (∅ x (A +𝑜 x) = suc y)))))
7311, 16, 21, 24, 72finds2 4251 . . . 4 (𝑏 𝜔 → (A 𝜔 → (A 𝑏x 𝜔 (∅ x (A +𝑜 x) = 𝑏))))
746, 73vtoclga 2596 . . 3 (B 𝜔 → (A 𝜔 → (A Bx 𝜔 (∅ x (A +𝑜 x) = B))))
7574impcom 116 . 2 ((A 𝜔 B 𝜔) → (A Bx 𝜔 (∅ x (A +𝑜 x) = B)))
76 peano1 4244 . . . . . . . . 9 𝜔
77 nnaord 5993 . . . . . . . . 9 ((∅ 𝜔 x 𝜔 A 𝜔) → (∅ x ↔ (A +𝑜 ∅) (A +𝑜 x)))
7876, 77mp3an1 1204 . . . . . . . 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 5968 . . . . . . . . 9 (A 𝜔 → (A +𝑜 ∅) = A)
8180adantr 261 . . . . . . . 8 ((A 𝜔 x 𝜔) → (A +𝑜 ∅) = A)
8281eleq1d 2088 . . . . . . 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 441 . . . . 5 ((A 𝜔 x 𝜔) → ((∅ x (A +𝑜 x) = B) ↔ (A (A +𝑜 x) (A +𝑜 x) = B)))
85 eleq2 2083 . . . . . 6 ((A +𝑜 x) = B → (A (A +𝑜 x) ↔ A B))
8685biimpac 282 . . . . 5 ((A (A +𝑜 x) (A +𝑜 x) = B) → A B)
8784, 86syl6bi 152 . . . 4 ((A 𝜔 x 𝜔) → ((∅ x (A +𝑜 x) = B) → A B))
8887rexlimdva 2411 . . 3 (A 𝜔 → (x 𝜔 (∅ x (A +𝑜 x) = B) → A B))
8988adantr 261 . 2 ((A 𝜔 B 𝜔) → (x 𝜔 (∅ x (A +𝑜 x) = B) → A B))
9075, 89impbid 120 1 ((A 𝜔 B 𝜔) → (A Bx 𝜔 (∅ x (A +𝑜 x) = B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 616   = wceq 1228   wcel 1374  wrex 2285  c0 3201  Oncon0 4049  suc csuc 4051  𝜔com 4240  (class class class)co 5436  1𝑜c1o 5909   +𝑜 coa 5913
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-1o 5916  df-oadd 5920
This theorem is referenced by:  nnawordex  6012  ltexpi  6197
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