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Theorem bj-inf2vnlem3 15402
Description: Lemma for bj-inf2vn 15404. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-inf2vnlem3.bd1 BOUNDED 𝐴
bj-inf2vnlem3.bd2 BOUNDED 𝑍
Assertion
Ref Expression
bj-inf2vnlem3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑍,𝑦

Proof of Theorem bj-inf2vnlem3
Dummy variables 𝑧 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem2 15401 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
2 bj-inf2vnlem3.bd1 . . . . . 6 BOUNDED 𝐴
32bdeli 15276 . . . . 5 BOUNDED 𝑧𝐴
4 bj-inf2vnlem3.bd2 . . . . . 6 BOUNDED 𝑍
54bdeli 15276 . . . . 5 BOUNDED 𝑧𝑍
63, 5ax-bdim 15244 . . . 4 BOUNDED (𝑧𝐴𝑧𝑍)
7 nfv 1539 . . . 4 𝑧(𝑡𝐴𝑡𝑍)
8 nfv 1539 . . . 4 𝑧(𝑢𝐴𝑢𝑍)
9 nfv 1539 . . . 4 𝑢(𝑧𝐴𝑧𝑍)
10 nfv 1539 . . . 4 𝑢(𝑡𝐴𝑡𝑍)
11 eleq1 2256 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝐴𝑡𝐴))
12 eleq1 2256 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝑍𝑡𝑍))
1311, 12imbi12d 234 . . . . 5 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) ↔ (𝑡𝐴𝑡𝑍)))
1413biimpd 144 . . . 4 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) → (𝑡𝐴𝑡𝑍)))
15 eleq1 2256 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝐴𝑢𝐴))
16 eleq1 2256 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝑍𝑢𝑍))
1715, 16imbi12d 234 . . . . 5 (𝑧 = 𝑢 → ((𝑧𝐴𝑧𝑍) ↔ (𝑢𝐴𝑢𝑍)))
1817biimprd 158 . . . 4 (𝑧 = 𝑢 → ((𝑢𝐴𝑢𝑍) → (𝑧𝐴𝑧𝑍)))
196, 7, 8, 9, 10, 14, 18bdsetindis 15399 . . 3 (∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)) → ∀𝑧(𝑧𝐴𝑧𝑍))
201, 19syl6 33 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑧(𝑧𝐴𝑧𝑍)))
21 dfss2 3168 . 2 (𝐴𝑍 ↔ ∀𝑧(𝑧𝐴𝑧𝑍))
2220, 21imbitrrdi 162 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 709  wal 1362   = wceq 1364  wcel 2164  wral 2472  wrex 2473  wss 3153  c0 3446  suc csuc 4394  BOUNDED wbdc 15270  Ind wind 15356
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bdim 15244  ax-bdsetind 15398
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-suc 4400  df-bdc 15271  df-bj-ind 15357
This theorem is referenced by:  bj-inf2vn  15404
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