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Theorem bj-nn0suc0 7319
Description: Constructive proof of a variant of nn0suc 4254. For a constructive proof of nn0suc 4254, see bj-nn0suc 7329. (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 2028 . . 3 (y = A → (y = ∅ ↔ A = ∅))
2 eqeq1 2028 . . . 4 (y = A → (y = suc xA = suc x))
32rexeqbi1dv 2492 . . 3 (y = A → (x y y = suc xx A A = suc x))
41, 3orbi12d 694 . 2 (y = A → ((y = ∅ x y y = suc x) ↔ (A = ∅ x A A = suc x)))
5 tru 1232 . . 3
6 a1tru 1244 . . . 4 ( ⊤ → ⊤ )
76rgenw 2354 . . 3 z 𝜔 ( ⊤ → ⊤ )
8 bdeq0 7241 . . . . 5 BOUNDED y = ∅
9 bdeqsuc 7255 . . . . . 6 BOUNDED y = suc x
109ax-bdex 7193 . . . . 5 BOUNDED x y y = suc x
118, 10ax-bdor 7190 . . . 4 BOUNDED (y = ∅ x y y = suc x)
12 nfv 1402 . . . 4 y
13 orc 620 . . . . 5 (y = ∅ → (y = ∅ x y y = suc x))
1413a1d 22 . . . 4 (y = ∅ → ( ⊤ → (y = ∅ x y y = suc x)))
15 a1tru 1244 . . . . 5 (¬ (y = z → ¬ (y = ∅ x y y = suc x)) → ⊤ )
1615expi 554 . . . 4 (y = z → ((y = ∅ x y y = suc x) → ⊤ ))
17 vex 2538 . . . . . . . . 9 z V
1817sucid 4103 . . . . . . . 8 z suc z
19 eleq2 2083 . . . . . . . 8 (y = suc z → (z yz suc z))
2018, 19mpbiri 157 . . . . . . 7 (y = suc zz y)
21 suceq 4088 . . . . . . . . 9 (x = z → suc x = suc z)
2221eqeq2d 2033 . . . . . . . 8 (x = z → (y = suc xy = suc z))
2322rspcev 2633 . . . . . . 7 ((z y y = suc z) → x y y = suc x)
2420, 23mpancom 401 . . . . . 6 (y = suc zx y y = suc x)
2524olcd 640 . . . . 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 7316 . . 3 (( ⊤ z 𝜔 ( ⊤ → ⊤ )) → y 𝜔 (y = ∅ x y y = suc x))
285, 7, 27mp2an 404 . 2 y 𝜔 (y = ∅ x y y = suc x)
294, 28vtoclri 2605 1 (A 𝜔 → (A = ∅ x A A = suc x))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 616   = wceq 1228  wtru 1229   wcel 1374  wral 2284  wrex 2285  c0 3201  suc csuc 4051  𝜔com 4240
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-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7187  ax-bdim 7188  ax-bdan 7189  ax-bdor 7190  ax-bdn 7191  ax-bdal 7192  ax-bdex 7193  ax-bdeq 7194  ax-bdel 7195  ax-bdsb 7196  ax-bdsep 7258  ax-infvn 7310
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7215  df-bj-ind 7297
This theorem is referenced by:  bj-nn0suc  7329
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