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Theorem nnaordex 6036
 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 2098 . . . . . 6 (𝑏 = B → (A 𝑏A B))
2 eqeq2 2046 . . . . . . . 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 2321 . . . . . 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 2098 . . . . . 6 (𝑏 = ∅ → (A 𝑏A ∅))
8 eqeq2 2046 . . . . . . . 8 (𝑏 = ∅ → ((A +𝑜 x) = 𝑏 ↔ (A +𝑜 x) = ∅))
98anbi2d 437 . . . . . . 7 (𝑏 = ∅ → ((∅ x (A +𝑜 x) = 𝑏) ↔ (∅ x (A +𝑜 x) = ∅)))
109rexbidv 2321 . . . . . 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 2098 . . . . . 6 (𝑏 = y → (A 𝑏A y))
13 eqeq2 2046 . . . . . . . 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 2321 . . . . . 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 2098 . . . . . 6 (𝑏 = suc y → (A 𝑏A suc y))
18 eqeq2 2046 . . . . . . . 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 2321 . . . . . 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 3222 . . . . . . 7 ¬ A
2322pm2.21i 574 . . . . . 6 (A ∅ → x 𝜔 (∅ x (A +𝑜 x) = ∅))
2423a1i 9 . . . . 5 (A 𝜔 → (A ∅ → x 𝜔 (∅ x (A +𝑜 x) = ∅)))
25 elsuci 4106 . . . . . . 7 (A suc y → (A y A = y))
26 simpr 103 . . . . . . . . 9 (((y 𝜔 A 𝜔) (A yx 𝜔 (∅ x (A +𝑜 x) = y))) → (A yx 𝜔 (∅ x (A +𝑜 x) = y)))
27 peano2 4261 . . . . . . . . . . . . . . 15 (x 𝜔 → suc x 𝜔)
2827ad2antlr 458 . . . . . . . . . . . . . 14 (((A 𝜔 x 𝜔) (∅ x (A +𝑜 x) = y)) → suc x 𝜔)
29 elelsuc 4112 . . . . . . . . . . . . . . . . 17 (∅ x → ∅ suc x)
3029a1i 9 . . . . . . . . . . . . . . . 16 ((A 𝜔 x 𝜔) → (∅ x → ∅ suc x))
31 nnasuc 5994 . . . . . . . . . . . . . . . . . 18 ((A 𝜔 x 𝜔) → (A +𝑜 suc x) = suc (A +𝑜 x))
32 suceq 4105 . . . . . . . . . . . . . . . . . 18 ((A +𝑜 x) = y → suc (A +𝑜 x) = suc y)
3331, 32sylan9eq 2089 . . . . . . . . . . . . . . . . 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 2098 . . . . . . . . . . . . . . . 16 (z = suc x → (∅ z ↔ ∅ suc x))
38 oveq2 5463 . . . . . . . . . . . . . . . . 17 (z = suc x → (A +𝑜 z) = (A +𝑜 suc x))
3938eqeq1d 2045 . . . . . . . . . . . . . . . 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 2650 . . . . . . . . . . . . . 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 2427 . . . . . . . . . . 11 (A 𝜔 → (x 𝜔 (∅ x (A +𝑜 x) = y) → z 𝜔 (∅ z (A +𝑜 z) = suc y)))
45 eleq2 2098 . . . . . . . . . . . . 13 (z = x → (∅ z ↔ ∅ x))
46 oveq2 5463 . . . . . . . . . . . . . 14 (z = x → (A +𝑜 z) = (A +𝑜 x))
4746eqeq1d 2045 . . . . . . . . . . . . 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 2528 . . . . . . . . . . 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) (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 5962 . . . . . . . . . . . 12 1𝑜
5453a1i 9 . . . . . . . . . . 11 ((A 𝜔 A = y) → ∅ 1𝑜)
55 nnon 4275 . . . . . . . . . . . . 13 (A 𝜔 → A On)
56 oa1suc 5986 . . . . . . . . . . . . 13 (A On → (A +𝑜 1𝑜) = suc A)
5755, 56syl 14 . . . . . . . . . . . 12 (A 𝜔 → (A +𝑜 1𝑜) = suc A)
58 suceq 4105 . . . . . . . . . . . 12 (A = y → suc A = suc y)
5957, 58sylan9eq 2089 . . . . . . . . . . 11 ((A 𝜔 A = y) → (A +𝑜 1𝑜) = suc y)
60 1onn 6029 . . . . . . . . . . . 12 1𝑜 𝜔
61 eleq2 2098 . . . . . . . . . . . . . 14 (x = 1𝑜 → (∅ x ↔ ∅ 1𝑜))
62 oveq2 5463 . . . . . . . . . . . . . . 15 (x = 1𝑜 → (A +𝑜 x) = (A +𝑜 1𝑜))
6362eqeq1d 2045 . . . . . . . . . . . . . 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 2650 . . . . . . . . . . . 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 = yx 𝜔 (∅ x (A +𝑜 x) = suc y)))
6968ad2antlr 458 . . . . . . . 8 (((y 𝜔 A 𝜔) (A yx 𝜔 (∅ x (A +𝑜 x) = y))) → (A = yx 𝜔 (∅ x (A +𝑜 x) = suc y)))
7052, 69jaod 636 . . . . . . 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 4267 . . . 4 (𝑏 𝜔 → (A 𝜔 → (A 𝑏x 𝜔 (∅ x (A +𝑜 x) = 𝑏))))
746, 73vtoclga 2613 . . 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 4260 . . . . . . . . 9 𝜔
77 nnaord 6018 . . . . . . . . 9 ((∅ 𝜔 x 𝜔 A 𝜔) → (∅ x ↔ (A +𝑜 ∅) (A +𝑜 x)))
7876, 77mp3an1 1218 . . . . . . . 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 5992 . . . . . . . . 9 (A 𝜔 → (A +𝑜 ∅) = A)
8180adantr 261 . . . . . . . 8 ((A 𝜔 x 𝜔) → (A +𝑜 ∅) = A)
8281eleq1d 2103 . . . . . . 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 2098 . . . . . 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 2427 . . 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 628   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  ∅c0 3218  Oncon0 4066  suc csuc 4068  𝜔com 4256  (class class class)co 5455  1𝑜c1o 5933   +𝑜 coa 5937 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944 This theorem is referenced by:  nnawordex  6037  ltexpi  6321
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