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Theorem bj-inf2vnlem4 13201
Description: Lemma for bj-inf2vn2 13203. (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 13199 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
2 nfv 1508 . . . 4 𝑧(𝑡𝐴𝑡𝑍)
3 nfv 1508 . . . 4 𝑧(𝑢𝐴𝑢𝑍)
4 nfv 1508 . . . 4 𝑢(𝑧𝐴𝑧𝑍)
5 nfv 1508 . . . 4 𝑢(𝑡𝐴𝑡𝑍)
6 eleq1 2202 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝐴𝑡𝐴))
7 eleq1 2202 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝑍𝑡𝑍))
86, 7imbi12d 233 . . . . 5 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) ↔ (𝑡𝐴𝑡𝑍)))
98biimpd 143 . . . 4 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) → (𝑡𝐴𝑡𝑍)))
10 eleq1 2202 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝐴𝑢𝐴))
11 eleq1 2202 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝑍𝑢𝑍))
1210, 11imbi12d 233 . . . . 5 (𝑧 = 𝑢 → ((𝑧𝐴𝑧𝑍) ↔ (𝑢𝐴𝑢𝑍)))
1312biimprd 157 . . . 4 (𝑧 = 𝑢 → ((𝑢𝐴𝑢𝑍) → (𝑧𝐴𝑧𝑍)))
142, 3, 4, 5, 9, 13setindis 13195 . . 3 (∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)) → ∀𝑧(𝑧𝐴𝑧𝑍))
151, 14syl6 33 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑧(𝑧𝐴𝑧𝑍)))
16 dfss2 3086 . 2 (𝐴𝑍 ↔ ∀𝑧(𝑧𝐴𝑧𝑍))
1715, 16syl6ibr 161 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 697  wal 1329   = wceq 1331  wcel 1480  wral 2416  wrex 2417  wss 3071  c0 3363  suc csuc 4287  Ind wind 13154
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-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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-suc 4293  df-bj-ind 13155
This theorem is referenced by:  bj-inf2vn2  13203
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