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Theorem bj-nn0suc0 14358
Description: Constructive proof of a variant of nn0suc 4600. For a constructive proof of nn0suc 4600, see bj-nn0suc 14372. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nn0suc0 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 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 2184 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2 eqeq1 2184 . . . 4 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
32rexeqbi1dv 2681 . . 3 (𝑦 = 𝐴 → (∃𝑥𝑦 𝑦 = suc 𝑥 ↔ ∃𝑥𝐴 𝐴 = suc 𝑥))
41, 3orbi12d 793 . 2 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥)))
5 tru 1357 . . 3
6 a1tru 1369 . . . 4 (⊤ → ⊤)
76rgenw 2532 . . 3 𝑧 ∈ ω (⊤ → ⊤)
8 bdeq0 14275 . . . . 5 BOUNDED 𝑦 = ∅
9 bdeqsuc 14289 . . . . . 6 BOUNDED 𝑦 = suc 𝑥
109ax-bdex 14227 . . . . 5 BOUNDED𝑥𝑦 𝑦 = suc 𝑥
118, 10ax-bdor 14224 . . . 4 BOUNDED (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
12 nfv 1528 . . . 4 𝑦
13 orc 712 . . . . 5 (𝑦 = ∅ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
1413a1d 22 . . . 4 (𝑦 = ∅ → (⊤ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)))
15 a1tru 1369 . . . . 5 (¬ (𝑦 = 𝑧 → ¬ (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)) → ⊤)
1615expi 638 . . . 4 (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) → ⊤))
17 vex 2740 . . . . . . . . 9 𝑧 ∈ V
1817sucid 4414 . . . . . . . 8 𝑧 ∈ suc 𝑧
19 eleq2 2241 . . . . . . . 8 (𝑦 = suc 𝑧 → (𝑧𝑦𝑧 ∈ suc 𝑧))
2018, 19mpbiri 168 . . . . . . 7 (𝑦 = suc 𝑧𝑧𝑦)
21 suceq 4399 . . . . . . . . 9 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2221eqeq2d 2189 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = suc 𝑥𝑦 = suc 𝑧))
2322rspcev 2841 . . . . . . 7 ((𝑧𝑦𝑦 = suc 𝑧) → ∃𝑥𝑦 𝑦 = suc 𝑥)
2420, 23mpancom 422 . . . . . 6 (𝑦 = suc 𝑧 → ∃𝑥𝑦 𝑦 = suc 𝑥)
2524olcd 734 . . . . 5 (𝑦 = suc 𝑧 → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
2625a1d 22 . . . 4 (𝑦 = suc 𝑧 → (⊤ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)))
2711, 12, 12, 12, 14, 16, 26bj-bdfindis 14355 . . 3 ((⊤ ∧ ∀𝑧 ∈ ω (⊤ → ⊤)) → ∀𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
285, 7, 27mp2an 426 . 2 𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
294, 28vtoclri 2812 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 708   = wceq 1353  wtru 1354  wcel 2148  wral 2455  wrex 2456  c0 3422  suc csuc 4362  ωcom 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4126  ax-pr 4206  ax-un 4430  ax-bd0 14221  ax-bdim 14222  ax-bdan 14223  ax-bdor 14224  ax-bdn 14225  ax-bdal 14226  ax-bdex 14227  ax-bdeq 14228  ax-bdel 14229  ax-bdsb 14230  ax-bdsep 14292  ax-infvn 14349
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-pr 3598  df-uni 3808  df-int 3843  df-suc 4368  df-iom 4587  df-bdc 14249  df-bj-ind 14335
This theorem is referenced by:  bj-nn0suc  14372
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