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Theorem bj-nn0suc0 9338
Description: Constructive proof of a variant of nn0suc 4270. For a constructive proof of nn0suc 4270, see bj-nn0suc 9348. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nn0suc0 (A 𝜔 → (A = ∅ x A A = suc x))
Distinct variable group:   x,A

Proof of Theorem bj-nn0suc0
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . 3 (y = A → (y = ∅ ↔ A = ∅))
2 eqeq1 2043 . . . 4 (y = A → (y = suc xA = suc x))
32rexeqbi1dv 2508 . . 3 (y = A → (x y y = suc xx A A = suc x))
41, 3orbi12d 706 . 2 (y = A → ((y = ∅ x y y = suc x) ↔ (A = ∅ x A A = suc x)))
5 tru 1246 . . 3
6 a1tru 1258 . . . 4 ( ⊤ → ⊤ )
76rgenw 2370 . . 3 z 𝜔 ( ⊤ → ⊤ )
8 bdeq0 9256 . . . . 5 BOUNDED y = ∅
9 bdeqsuc 9270 . . . . . 6 BOUNDED y = suc x
109ax-bdex 9208 . . . . 5 BOUNDED x y y = suc x
118, 10ax-bdor 9205 . . . 4 BOUNDED (y = ∅ x y y = suc x)
12 nfv 1418 . . . 4 y
13 orc 632 . . . . 5 (y = ∅ → (y = ∅ x y y = suc x))
1413a1d 22 . . . 4 (y = ∅ → ( ⊤ → (y = ∅ x y y = suc x)))
15 a1tru 1258 . . . . 5 (¬ (y = z → ¬ (y = ∅ x y y = suc x)) → ⊤ )
1615expi 566 . . . 4 (y = z → ((y = ∅ x y y = suc x) → ⊤ ))
17 vex 2554 . . . . . . . . 9 z V
1817sucid 4120 . . . . . . . 8 z suc z
19 eleq2 2098 . . . . . . . 8 (y = suc z → (z yz suc z))
2018, 19mpbiri 157 . . . . . . 7 (y = suc zz y)
21 suceq 4105 . . . . . . . . 9 (x = z → suc x = suc z)
2221eqeq2d 2048 . . . . . . . 8 (x = z → (y = suc xy = suc z))
2322rspcev 2650 . . . . . . 7 ((z y y = suc z) → x y y = suc x)
2420, 23mpancom 399 . . . . . 6 (y = suc zx y y = suc x)
2524olcd 652 . . . . 5 (y = suc z → (y = ∅ x y y = suc x))
2625a1d 22 . . . 4 (y = suc z → ( ⊤ → (y = ∅ x y y = suc x)))
2711, 12, 12, 12, 14, 16, 26bj-bdfindis 9335 . . 3 (( ⊤ z 𝜔 ( ⊤ → ⊤ )) → y 𝜔 (y = ∅ x y y = suc x))
285, 7, 27mp2an 402 . 2 y 𝜔 (y = ∅ x y y = suc x)
294, 28vtoclri 2622 1 (A 𝜔 → (A = ∅ x A A = suc x))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 628   = wceq 1242  wtru 1243   wcel 1390  wral 2300  wrex 2301  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 9202  ax-bdim 9203  ax-bdan 9204  ax-bdor 9205  ax-bdn 9206  ax-bdal 9207  ax-bdex 9208  ax-bdeq 9209  ax-bdel 9210  ax-bdsb 9211  ax-bdsep 9273  ax-infvn 9329
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  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 9230  df-bj-ind 9316
This theorem is referenced by:  bj-nn0suc  9348
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