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Theorem bj-nn0suc0 12553
Description: Constructive proof of a variant of nn0suc 4447. For a constructive proof of nn0suc 4447, see bj-nn0suc 12567. (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 2101 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2 eqeq1 2101 . . . 4 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
32rexeqbi1dv 2585 . . 3 (𝑦 = 𝐴 → (∃𝑥𝑦 𝑦 = suc 𝑥 ↔ ∃𝑥𝐴 𝐴 = suc 𝑥))
41, 3orbi12d 745 . 2 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥)))
5 tru 1300 . . 3
6 a1tru 1312 . . . 4 (⊤ → ⊤)
76rgenw 2441 . . 3 𝑧 ∈ ω (⊤ → ⊤)
8 bdeq0 12466 . . . . 5 BOUNDED 𝑦 = ∅
9 bdeqsuc 12480 . . . . . 6 BOUNDED 𝑦 = suc 𝑥
109ax-bdex 12418 . . . . 5 BOUNDED𝑥𝑦 𝑦 = suc 𝑥
118, 10ax-bdor 12415 . . . 4 BOUNDED (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
12 nfv 1473 . . . 4 𝑦
13 orc 671 . . . . 5 (𝑦 = ∅ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
1413a1d 22 . . . 4 (𝑦 = ∅ → (⊤ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)))
15 a1tru 1312 . . . . 5 (¬ (𝑦 = 𝑧 → ¬ (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)) → ⊤)
1615expi 605 . . . 4 (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) → ⊤))
17 vex 2636 . . . . . . . . 9 𝑧 ∈ V
1817sucid 4268 . . . . . . . 8 𝑧 ∈ suc 𝑧
19 eleq2 2158 . . . . . . . 8 (𝑦 = suc 𝑧 → (𝑧𝑦𝑧 ∈ suc 𝑧))
2018, 19mpbiri 167 . . . . . . 7 (𝑦 = suc 𝑧𝑧𝑦)
21 suceq 4253 . . . . . . . . 9 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2221eqeq2d 2106 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = suc 𝑥𝑦 = suc 𝑧))
2322rspcev 2736 . . . . . . 7 ((𝑧𝑦𝑦 = suc 𝑧) → ∃𝑥𝑦 𝑦 = suc 𝑥)
2420, 23mpancom 414 . . . . . 6 (𝑦 = suc 𝑧 → ∃𝑥𝑦 𝑦 = suc 𝑥)
2524olcd 691 . . . . 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 12550 . . 3 ((⊤ ∧ ∀𝑧 ∈ ω (⊤ → ⊤)) → ∀𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
285, 7, 27mp2an 418 . 2 𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
294, 28vtoclri 2708 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 667   = wceq 1296  wtru 1297  wcel 1445  wral 2370  wrex 2371  c0 3302  suc csuc 4216  ωcom 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-nul 3986  ax-pr 4060  ax-un 4284  ax-bd0 12412  ax-bdim 12413  ax-bdan 12414  ax-bdor 12415  ax-bdn 12416  ax-bdal 12417  ax-bdex 12418  ax-bdeq 12419  ax-bdel 12420  ax-bdsb 12421  ax-bdsep 12483  ax-infvn 12544
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711  df-suc 4222  df-iom 4434  df-bdc 12440  df-bj-ind 12530
This theorem is referenced by:  bj-nn0suc  12567
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