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Theorem bj-nn0suc0 14973
Description: Constructive proof of a variant of nn0suc 4615. For a constructive proof of nn0suc 4615, see bj-nn0suc 14987. (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 2194 . . 3 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2 eqeq1 2194 . . . 4 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
32rexeqbi1dv 2692 . . 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 1367 . . 3
6 trud 1379 . . . 4 (⊤ → ⊤)
76rgenw 2542 . . 3 𝑧 ∈ ω (⊤ → ⊤)
8 bdeq0 14890 . . . . 5 BOUNDED 𝑦 = ∅
9 bdeqsuc 14904 . . . . . 6 BOUNDED 𝑦 = suc 𝑥
109ax-bdex 14842 . . . . 5 BOUNDED𝑥𝑦 𝑦 = suc 𝑥
118, 10ax-bdor 14839 . . . 4 BOUNDED (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
12 nfv 1538 . . . 4 𝑦
13 orc 713 . . . . 5 (𝑦 = ∅ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
1413a1d 22 . . . 4 (𝑦 = ∅ → (⊤ → (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)))
15 trud 1379 . . . . 5 (¬ (𝑦 = 𝑧 → ¬ (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)) → ⊤)
1615expi 639 . . . 4 (𝑦 = 𝑧 → ((𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥) → ⊤))
17 vex 2752 . . . . . . . . 9 𝑧 ∈ V
1817sucid 4429 . . . . . . . 8 𝑧 ∈ suc 𝑧
19 eleq2 2251 . . . . . . . 8 (𝑦 = suc 𝑧 → (𝑧𝑦𝑧 ∈ suc 𝑧))
2018, 19mpbiri 168 . . . . . . 7 (𝑦 = suc 𝑧𝑧𝑦)
21 suceq 4414 . . . . . . . . 9 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
2221eqeq2d 2199 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 = suc 𝑥𝑦 = suc 𝑧))
2322rspcev 2853 . . . . . . 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 14970 . . 3 ((⊤ ∧ ∀𝑧 ∈ ω (⊤ → ⊤)) → ∀𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥))
285, 7, 27mp2an 426 . 2 𝑦 ∈ ω (𝑦 = ∅ ∨ ∃𝑥𝑦 𝑦 = suc 𝑥)
294, 28vtoclri 2824 1 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 709   = wceq 1363  wtru 1364  wcel 2158  wral 2465  wrex 2466  c0 3434  suc csuc 4377  ωcom 4601
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-nul 4141  ax-pr 4221  ax-un 4445  ax-bd0 14836  ax-bdim 14837  ax-bdan 14838  ax-bdor 14839  ax-bdn 14840  ax-bdal 14841  ax-bdex 14842  ax-bdeq 14843  ax-bdel 14844  ax-bdsb 14845  ax-bdsep 14907  ax-infvn 14964
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-suc 4383  df-iom 4602  df-bdc 14864  df-bj-ind 14950
This theorem is referenced by:  bj-nn0suc  14987
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