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Theorem bj-inf2vnlem1 14993
Description: Lemma for bj-inf2vn 14997. 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 biimpr 130 . . . . 5 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴))
2 jaob 711 . . . . . 6 (((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴) ↔ ((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)))
32biimpi 120 . . . . 5 (((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → 𝑥𝐴) → ((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)))
4 simpl 109 . . . . . 6 (((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)) → (𝑥 = ∅ → 𝑥𝐴))
5 eleq1 2250 . . . . . 6 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ∈ 𝐴))
64, 5mpbidi 151 . . . . 5 (((𝑥 = ∅ → 𝑥𝐴) ∧ (∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴)) → (𝑥 = ∅ → ∅ ∈ 𝐴))
71, 3, 63syl 17 . . . 4 ((𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → (𝑥 = ∅ → ∅ ∈ 𝐴))
87alimi 1465 . . 3 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴))
9 exim 1609 . . 3 (∀𝑥(𝑥 = ∅ → ∅ ∈ 𝐴) → (∃𝑥 𝑥 = ∅ → ∃𝑥∅ ∈ 𝐴))
10 0ex 4142 . . . . . 6 ∅ ∈ V
1110isseti 2757 . . . . 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 14785 . . . 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 114 . . . . . 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 1465 . . . 4 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑥(∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴))
20 eqid 2187 . . . . 5 suc 𝑧 = suc 𝑧
21 suceq 4414 . . . . . . 7 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
2221eqeq2d 2199 . . . . . 6 (𝑦 = 𝑧 → (suc 𝑧 = suc 𝑦 ↔ suc 𝑧 = suc 𝑧))
2322rspcev 2853 . . . . 5 ((𝑧𝐴 ∧ suc 𝑧 = suc 𝑧) → ∃𝑦𝐴 suc 𝑧 = suc 𝑦)
2420, 23mpan2 425 . . . 4 (𝑧𝐴 → ∃𝑦𝐴 suc 𝑧 = suc 𝑦)
25 vex 2752 . . . . . 6 𝑧 ∈ V
2625bj-sucex 14946 . . . . 5 suc 𝑧 ∈ V
27 eqeq1 2194 . . . . . . 7 (𝑥 = suc 𝑧 → (𝑥 = suc 𝑦 ↔ suc 𝑧 = suc 𝑦))
2827rexbidv 2488 . . . . . 6 (𝑥 = suc 𝑧 → (∃𝑦𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦𝐴 suc 𝑧 = suc 𝑦))
29 eleq1 2250 . . . . . 6 (𝑥 = suc 𝑧 → (𝑥𝐴 ↔ suc 𝑧𝐴))
3028, 29imbi12d 234 . . . . 5 (𝑥 = suc 𝑧 → ((∃𝑦𝐴 𝑥 = suc 𝑦𝑥𝐴) ↔ (∃𝑦𝐴 suc 𝑧 = suc 𝑦 → suc 𝑧𝐴)))
3126, 30spcv 2843 . . . 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 2559 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → ∀𝑧𝐴 suc 𝑧𝐴)
34 df-bj-ind 14950 . 2 (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑧𝐴 suc 𝑧𝐴))
3516, 33, 34sylanbrc 417 1 (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709  wal 1361   = wceq 1363  wex 1502  wcel 2158  wral 2465  wrex 2466  c0 3434  suc csuc 4377  Ind wind 14949
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-bdor 14839  ax-bdex 14842  ax-bdeq 14843  ax-bdel 14844  ax-bdsb 14845  ax-bdsep 14907
This theorem depends on definitions:  df-bi 117  df-tru 1366  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-v 2751  df-dif 3143  df-un 3145  df-nul 3435  df-sn 3610  df-pr 3611  df-uni 3822  df-suc 4383  df-bdc 14864  df-bj-ind 14950
This theorem is referenced by:  bj-inf2vn  14997  bj-inf2vn2  14998
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