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Theorem bj-inf2vnlem1 13157
Description: Lemma for bj-inf2vn 13161. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inf2vnlem1 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem bj-inf2vnlem1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bi2 129 . . . . 5 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴))
2 jaob 699 . . . . . 6 (((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴) ↔ ((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)))
32biimpi 119 . . . . 5 (((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴) → ((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)))
4 simpl 108 . . . . . 6 (((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)) → (𝑥 = ∅ → 𝑥𝐴))
5 eleq1 2200 . . . . . 6 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ∈ 𝐴))
64, 5mpbidi 150 . . . . 5 (((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)) → (𝑥 = ∅ → ∅ ∈ 𝐴))
71, 3, 63syl 17 . . . 4 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (𝑥 = ∅ → ∅ ∈ 𝐴))
87alimi 1431 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴))
9 exim 1578 . . 3 (∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴) → (∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴))
10 0ex 4050 . . . . . 6 ∅ ∈ V
1110isseti 2689 . . . . 5 𝑥 𝑥 = ∅
12 pm2.27 40 . . . . 5 (∃𝑥 𝑥 = ∅ → ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∃𝑥∅ ∈ 𝐴))
1311, 12ax-mp 5 . . . 4 ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∃𝑥∅ ∈ 𝐴)
14 bj-ex 12958 . . . 4 (∃𝑥∅ ∈ 𝐴 → ∅ ∈ 𝐴)
1513, 14syl 14 . . 3 ((∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴) → ∅ ∈ 𝐴)
168, 9, 153syl 17 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∅ ∈ 𝐴)
173simprd 113 . . . . . 6 (((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴) → (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴))
181, 17syl 14 . . . . 5 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴))
1918alimi 1431 . . . 4 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴))
20 eqid 2137 . . . . 5 suc 𝑧 = suc 𝑧
21 suceq 4319 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
2221eqeq2d 2149 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑧 = suc 𝑦 ↔ suc 𝑧 = suc 𝑧))
2322rspcev 2784 . . . . 5 ((𝑧𝐴 ∧ suc 𝑧 = suc 𝑧) → ∃𝑦𝐴 suc 𝑧 = suc 𝑦)
2420, 23mpan2 421 . . . 4 (𝑧𝐴 → ∃𝑦𝐴 suc 𝑧 = suc 𝑦)
25 vex 2684 . . . . . 6 𝑧 ∈ V
2625bj-sucex 13110 . . . . 5 suc 𝑧 ∈ V
27 eqeq1 2144 . . . . . . 7 (𝑥 = suc 𝑧 → (𝑥 = suc 𝑦 ↔ suc 𝑧 = suc 𝑦))
2827rexbidv 2436 . . . . . 6 (𝑥 = suc 𝑧 → (∃𝑦𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦𝐴 suc 𝑧 = suc 𝑦))
29 eleq1 2200 . . . . . 6 (𝑥 = suc 𝑧 → (𝑥𝐴 ↔ suc 𝑧𝐴))
3028, 29imbi12d 233 . . . . 5 (𝑥 = suc 𝑧 → ((∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴) ↔ (∃𝑦𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧𝐴)))
3126, 30spcv 2774 . . . 4 (∀𝑥(∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴) → (∃𝑦𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧𝐴))
3219, 24, 31syl2im 38 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (𝑧𝐴 → suc 𝑧𝐴))
3332ralrimiv 2502 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑧𝐴 suc 𝑧𝐴)
34 df-bj-ind 13114 . 2 (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑧𝐴 suc 𝑧𝐴))
3516, 33, 34sylanbrc 413 1 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 697  wal 1329   = wceq 1331  wex 1468  wcel 1480  wral 2414  wrex 2415  c0 3358  suc csuc 4282  Ind wind 13113
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049  ax-pr 4126  ax-un 4350  ax-bd0 13000  ax-bdor 13003  ax-bdex 13006  ax-bdeq 13007  ax-bdel 13008  ax-bdsb 13009  ax-bdsep 13071
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-nul 3359  df-sn 3528  df-pr 3529  df-uni 3732  df-suc 4288  df-bdc 13028  df-bj-ind 13114
This theorem is referenced by:  bj-inf2vn  13161  bj-inf2vn2  13162
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