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Theorem bj-nn0suc0 15105
Description: Constructive proof of a variant of nn0suc 4618. For a constructive proof of nn0suc 4618, see bj-nn0suc 15119. (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 2196 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2 eqeq1 2196 . . . 4 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
32rexeqbi1dv 2695 . . 3 (𝑦 = 𝐴 → (∃𝑥𝑦 𝑦 = suc 𝑥 ↔ ∃𝑥𝐴 𝐴 = suc 𝑥))
41, 3orbi12d 794 . 2 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥)))
5 tru 1368 . . 3
6 trud 1380 . . . 4 (⊤ → ⊤)
76rgenw 2545 . . 3 𝑧 ∈ ω (⊤ → ⊤)
8 bdeq0 15022 . . . . 5 BOUNDED 𝑦 = ∅
9 bdeqsuc 15036 . . . . . 6 BOUNDED 𝑦 = suc 𝑥
109ax-bdex 14974 . . . . 5 BOUNDED𝑥𝑦 𝑦 = suc 𝑥
118, 10ax-bdor 14971 . . . 4 BOUNDED (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
12 nfv 1539 . . . 4 𝑦
13 orc 713 . . . . 5 (𝑦 = ∅ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
1413a1d 22 . . . 4 (𝑦 = ∅ → (⊤ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)))
15 trud 1380 . . . . 5 (¬ (𝑦 = 𝑧 → ¬ (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)) → ⊤)
1615expi 639 . . . 4 (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) → ⊤))
17 vex 2755 . . . . . . . . 9 𝑧 ∈ V
1817sucid 4432 . . . . . . . 8 𝑧 ∈ suc 𝑧
19 eleq2 2253 . . . . . . . 8 (𝑦 = suc 𝑧 → (𝑧𝑦𝑧 ∈ suc 𝑧))
2018, 19mpbiri 168 . . . . . . 7 (𝑦 = suc 𝑧𝑧𝑦)
21 suceq 4417 . . . . . . . . 9 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2221eqeq2d 2201 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = suc 𝑥𝑦 = suc 𝑧))
2322rspcev 2856 . . . . . . 7 ((𝑧𝑦𝑦 = suc 𝑧) → ∃𝑥𝑦 𝑦 = suc 𝑥)
2420, 23mpancom 422 . . . . . 6 (𝑦 = suc 𝑧 → ∃𝑥𝑦 𝑦 = suc 𝑥)
2524olcd 735 . . . . 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 15102 . . 3 ((⊤ ∧ ∀𝑧 ∈ ω (⊤ → ⊤)) → ∀𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
285, 7, 27mp2an 426 . 2 𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
294, 28vtoclri 2827 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 709   = wceq 1364  wtru 1365  wcel 2160  wral 2468  wrex 2469  c0 3437  suc csuc 4380  ωcom 4604
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4224  ax-un 4448  ax-bd0 14968  ax-bdim 14969  ax-bdan 14970  ax-bdor 14971  ax-bdn 14972  ax-bdal 14973  ax-bdex 14974  ax-bdeq 14975  ax-bdel 14976  ax-bdsb 14977  ax-bdsep 15039  ax-infvn 15096
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4386  df-iom 4605  df-bdc 14996  df-bj-ind 15082
This theorem is referenced by:  bj-nn0suc  15119
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