Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-nn0suc0 GIF version

Theorem bj-nn0suc0 13075
Description: Constructive proof of a variant of nn0suc 4488. For a constructive proof of nn0suc 4488, see bj-nn0suc 13089. (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 2124 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2 eqeq1 2124 . . . 4 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
32rexeqbi1dv 2612 . . 3 (𝑦 = 𝐴 → (∃𝑥𝑦 𝑦 = suc 𝑥 ↔ ∃𝑥𝐴 𝐴 = suc 𝑥))
41, 3orbi12d 767 . 2 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) ↔ (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥)))
5 tru 1320 . . 3
6 a1tru 1332 . . . 4 (⊤ → ⊤)
76rgenw 2464 . . 3 𝑧 ∈ ω (⊤ → ⊤)
8 bdeq0 12992 . . . . 5 BOUNDED 𝑦 = ∅
9 bdeqsuc 13006 . . . . . 6 BOUNDED 𝑦 = suc 𝑥
109ax-bdex 12944 . . . . 5 BOUNDED𝑥𝑦 𝑦 = suc 𝑥
118, 10ax-bdor 12941 . . . 4 BOUNDED (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
12 nfv 1493 . . . 4 𝑦
13 orc 686 . . . . 5 (𝑦 = ∅ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
1413a1d 22 . . . 4 (𝑦 = ∅ → (⊤ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)))
15 a1tru 1332 . . . . 5 (¬ (𝑦 = 𝑧 → ¬ (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)) → ⊤)
1615expi 612 . . . 4 (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) → ⊤))
17 vex 2663 . . . . . . . . 9 𝑧 ∈ V
1817sucid 4309 . . . . . . . 8 𝑧 ∈ suc 𝑧
19 eleq2 2181 . . . . . . . 8 (𝑦 = suc 𝑧 → (𝑧𝑦𝑧 ∈ suc 𝑧))
2018, 19mpbiri 167 . . . . . . 7 (𝑦 = suc 𝑧𝑧𝑦)
21 suceq 4294 . . . . . . . . 9 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2221eqeq2d 2129 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = suc 𝑥𝑦 = suc 𝑧))
2322rspcev 2763 . . . . . . 7 ((𝑧𝑦𝑦 = suc 𝑧) → ∃𝑥𝑦 𝑦 = suc 𝑥)
2420, 23mpancom 418 . . . . . 6 (𝑦 = suc 𝑧 → ∃𝑥𝑦 𝑦 = suc 𝑥)
2524olcd 708 . . . . 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 13072 . . 3 ((⊤ ∧ ∀𝑧 ∈ ω (⊤ → ⊤)) → ∀𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
285, 7, 27mp2an 422 . 2 𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
294, 28vtoclri 2735 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 682   = wceq 1316  wtru 1317  wcel 1465  wral 2393  wrex 2394  c0 3333  suc csuc 4257  ωcom 4474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdim 12939  ax-bdan 12940  ax-bdor 12941  ax-bdn 12942  ax-bdal 12943  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009  ax-infvn 13066
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by:  bj-nn0suc  13089
  Copyright terms: Public domain W3C validator