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Theorem bj-nn0sucALT 9408
Description: Alternate proof of bj-nn0suc 9394, also constructive but from ax-inf2 9406, hence requiring ax-bdsetind 9398. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-nn0sucALT (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))
Distinct variable group:   x,A

Proof of Theorem bj-nn0sucALT
Dummy variables 𝑎 y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-inf2 9406 . . 3 𝑎y(y 𝑎 ↔ (y = ∅ z 𝑎 y = suc z))
2 vex 2554 . . . . 5 𝑎 V
3 bdcv 9283 . . . . . 6 BOUNDED 𝑎
43bj-inf2vn 9404 . . . . 5 (𝑎 V → (y(y 𝑎 ↔ (y = ∅ z 𝑎 y = suc z)) → 𝑎 = 𝜔))
52, 4ax-mp 7 . . . 4 (y(y 𝑎 ↔ (y = ∅ z 𝑎 y = suc z)) → 𝑎 = 𝜔)
6 eleq2 2098 . . . . . . 7 (𝑎 = 𝜔 → (y 𝑎y 𝜔))
7 rexeq 2500 . . . . . . . 8 (𝑎 = 𝜔 → (z 𝑎 y = suc zz 𝜔 y = suc z))
87orbi2d 703 . . . . . . 7 (𝑎 = 𝜔 → ((y = ∅ z 𝑎 y = suc z) ↔ (y = ∅ z 𝜔 y = suc z)))
96, 8bibi12d 224 . . . . . 6 (𝑎 = 𝜔 → ((y 𝑎 ↔ (y = ∅ z 𝑎 y = suc z)) ↔ (y 𝜔 ↔ (y = ∅ z 𝜔 y = suc z))))
109albidv 1702 . . . . 5 (𝑎 = 𝜔 → (y(y 𝑎 ↔ (y = ∅ z 𝑎 y = suc z)) ↔ y(y 𝜔 ↔ (y = ∅ z 𝜔 y = suc z))))
11 nfcv 2175 . . . . . . . 8 yA
12 nfv 1418 . . . . . . . 8 y(A 𝜔 → (A = ∅ x 𝜔 A = suc x))
13 eleq1 2097 . . . . . . . . . 10 (y = A → (y 𝜔 ↔ A 𝜔))
14 eqeq1 2043 . . . . . . . . . . 11 (y = A → (y = ∅ ↔ A = ∅))
15 suceq 4105 . . . . . . . . . . . . . 14 (z = x → suc z = suc x)
1615eqeq2d 2048 . . . . . . . . . . . . 13 (z = x → (y = suc zy = suc x))
1716cbvrexv 2528 . . . . . . . . . . . 12 (z 𝜔 y = suc zx 𝜔 y = suc x)
18 eqeq1 2043 . . . . . . . . . . . . 13 (y = A → (y = suc xA = suc x))
1918rexbidv 2321 . . . . . . . . . . . 12 (y = A → (x 𝜔 y = suc xx 𝜔 A = suc x))
2017, 19syl5bb 181 . . . . . . . . . . 11 (y = A → (z 𝜔 y = suc zx 𝜔 A = suc x))
2114, 20orbi12d 706 . . . . . . . . . 10 (y = A → ((y = ∅ z 𝜔 y = suc z) ↔ (A = ∅ x 𝜔 A = suc x)))
2213, 21bibi12d 224 . . . . . . . . 9 (y = A → ((y 𝜔 ↔ (y = ∅ z 𝜔 y = suc z)) ↔ (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))))
23 bi1 111 . . . . . . . . 9 ((A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x)) → (A 𝜔 → (A = ∅ x 𝜔 A = suc x)))
2422, 23syl6bi 152 . . . . . . . 8 (y = A → ((y 𝜔 ↔ (y = ∅ z 𝜔 y = suc z)) → (A 𝜔 → (A = ∅ x 𝜔 A = suc x))))
2511, 12, 24spcimgf 2627 . . . . . . 7 (A 𝜔 → (y(y 𝜔 ↔ (y = ∅ z 𝜔 y = suc z)) → (A 𝜔 → (A = ∅ x 𝜔 A = suc x))))
2625pm2.43b 46 . . . . . 6 (y(y 𝜔 ↔ (y = ∅ z 𝜔 y = suc z)) → (A 𝜔 → (A = ∅ x 𝜔 A = suc x)))
27 peano1 4260 . . . . . . . 8 𝜔
28 eleq1 2097 . . . . . . . 8 (A = ∅ → (A 𝜔 ↔ ∅ 𝜔))
2927, 28mpbiri 157 . . . . . . 7 (A = ∅ → A 𝜔)
30 bj-peano2 9373 . . . . . . . . 9 (x 𝜔 → suc x 𝜔)
31 eleq1a 2106 . . . . . . . . . 10 (suc x 𝜔 → (A = suc xA 𝜔))
3231imp 115 . . . . . . . . 9 ((suc x 𝜔 A = suc x) → A 𝜔)
3330, 32sylan 267 . . . . . . . 8 ((x 𝜔 A = suc x) → A 𝜔)
3433rexlimiva 2422 . . . . . . 7 (x 𝜔 A = suc xA 𝜔)
3529, 34jaoi 635 . . . . . 6 ((A = ∅ x 𝜔 A = suc x) → A 𝜔)
3626, 35impbid1 130 . . . . 5 (y(y 𝜔 ↔ (y = ∅ z 𝜔 y = suc z)) → (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x)))
3710, 36syl6bi 152 . . . 4 (𝑎 = 𝜔 → (y(y 𝑎 ↔ (y = ∅ z 𝑎 y = suc z)) → (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))))
385, 37mpcom 32 . . 3 (y(y 𝑎 ↔ (y = ∅ z 𝑎 y = suc z)) → (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x)))
391, 38eximii 1490 . 2 𝑎(A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))
40 bj-ex 9217 . 2 (𝑎(A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x)) → (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x)))
4139, 40ax-mp 7 1 (A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wo 628  wal 1240   = wceq 1242  wex 1378   wcel 1390  wrex 2301  Vcvv 2551  c0 3218  suc csuc 4068  𝜔com 4256
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-bnd 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-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9248  ax-bdim 9249  ax-bdor 9251  ax-bdex 9254  ax-bdeq 9255  ax-bdel 9256  ax-bdsb 9257  ax-bdsep 9319  ax-bdsetind 9398  ax-inf2 9406
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9276  df-bj-ind 9362
This theorem is referenced by: (None)
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