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Theorem bj-inf2vnlem4 10098
Description: Lemma for bj-inf2vn2 10100. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inf2vnlem4 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑍,𝑦

Proof of Theorem bj-inf2vnlem4
Dummy variables 𝑧 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem2 10096 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
2 nfv 1421 . . . 4 𝑧(𝑡𝐴𝑡𝑍)
3 nfv 1421 . . . 4 𝑧(𝑢𝐴𝑢𝑍)
4 nfv 1421 . . . 4 𝑢(𝑧𝐴𝑧𝑍)
5 nfv 1421 . . . 4 𝑢(𝑡𝐴𝑡𝑍)
6 eleq1 2100 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝐴𝑡𝐴))
7 eleq1 2100 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝑍𝑡𝑍))
86, 7imbi12d 223 . . . . 5 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) ↔ (𝑡𝐴𝑡𝑍)))
98biimpd 132 . . . 4 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) → (𝑡𝐴𝑡𝑍)))
10 eleq1 2100 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝐴𝑢𝐴))
11 eleq1 2100 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝑍𝑢𝑍))
1210, 11imbi12d 223 . . . . 5 (𝑧 = 𝑢 → ((𝑧𝐴𝑧𝑍) ↔ (𝑢𝐴𝑢𝑍)))
1312biimprd 147 . . . 4 (𝑧 = 𝑢 → ((𝑢𝐴𝑢𝑍) → (𝑧𝐴𝑧𝑍)))
142, 3, 4, 5, 9, 13setindis 10092 . . 3 (∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)) → ∀𝑧(𝑧𝐴𝑧𝑍))
151, 14syl6 29 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑧(𝑧𝐴𝑧𝑍)))
16 dfss2 2934 . 2 (𝐴𝑍 ↔ ∀𝑧(𝑧𝐴𝑧𝑍))
1715, 16syl6ibr 151 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 629  wal 1241   = wceq 1243  wcel 1393  wral 2306  wrex 2307  wss 2917  c0 3224  suc csuc 4102  Ind wind 10050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-suc 4108  df-bj-ind 10051
This theorem is referenced by:  bj-inf2vn2  10100
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