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Theorem bj-inf2vnlem3 12972
Description: Lemma for bj-inf2vn 12974. (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 12971 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
2 bj-inf2vnlem3.bd1 . . . . . 6 BOUNDED 𝐴
32bdeli 12846 . . . . 5 BOUNDED 𝑧𝐴
4 bj-inf2vnlem3.bd2 . . . . . 6 BOUNDED 𝑍
54bdeli 12846 . . . . 5 BOUNDED 𝑧𝑍
63, 5ax-bdim 12814 . . . 4 BOUNDED (𝑧𝐴𝑧𝑍)
7 nfv 1491 . . . 4 𝑧(𝑡𝐴𝑡𝑍)
8 nfv 1491 . . . 4 𝑧(𝑢𝐴𝑢𝑍)
9 nfv 1491 . . . 4 𝑢(𝑧𝐴𝑧𝑍)
10 nfv 1491 . . . 4 𝑢(𝑡𝐴𝑡𝑍)
11 eleq1 2178 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝐴𝑡𝐴))
12 eleq1 2178 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝑍𝑡𝑍))
1311, 12imbi12d 233 . . . . 5 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) ↔ (𝑡𝐴𝑡𝑍)))
1413biimpd 143 . . . 4 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) → (𝑡𝐴𝑡𝑍)))
15 eleq1 2178 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝐴𝑢𝐴))
16 eleq1 2178 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝑍𝑢𝑍))
1715, 16imbi12d 233 . . . . 5 (𝑧 = 𝑢 → ((𝑧𝐴𝑧𝑍) ↔ (𝑢𝐴𝑢𝑍)))
1817biimprd 157 . . . 4 (𝑧 = 𝑢 → ((𝑢𝐴𝑢𝑍) → (𝑧𝐴𝑧𝑍)))
196, 7, 8, 9, 10, 14, 18bdsetindis 12969 . . 3 (∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)) → ∀𝑧(𝑧𝐴𝑧𝑍))
201, 19syl6 33 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑧(𝑧𝐴𝑧𝑍)))
21 dfss2 3054 . 2 (𝐴𝑍 ↔ ∀𝑧(𝑧𝐴𝑧𝑍))
2220, 21syl6ibr 161 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 680  wal 1312   = wceq 1314  wcel 1463  wral 2391  wrex 2392  wss 3039  c0 3331  suc csuc 4255  BOUNDED wbdc 12840  Ind wind 12926
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bdim 12814  ax-bdsetind 12968
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-suc 4261  df-bdc 12841  df-bj-ind 12927
This theorem is referenced by:  bj-inf2vn  12974
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