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Theorem bj-inf2vnlem3 10456
Description: Lemma for bj-inf2vn 10458. (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 10455 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
2 bj-inf2vnlem3.bd1 . . . . . 6 BOUNDED 𝐴
32bdeli 10325 . . . . 5 BOUNDED 𝑧𝐴
4 bj-inf2vnlem3.bd2 . . . . . 6 BOUNDED 𝑍
54bdeli 10325 . . . . 5 BOUNDED 𝑧𝑍
63, 5ax-bdim 10293 . . . 4 BOUNDED (𝑧𝐴𝑧𝑍)
7 nfv 1437 . . . 4 𝑧(𝑡𝐴𝑡𝑍)
8 nfv 1437 . . . 4 𝑧(𝑢𝐴𝑢𝑍)
9 nfv 1437 . . . 4 𝑢(𝑧𝐴𝑧𝑍)
10 nfv 1437 . . . 4 𝑢(𝑡𝐴𝑡𝑍)
11 eleq1 2116 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝐴𝑡𝐴))
12 eleq1 2116 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝑍𝑡𝑍))
1311, 12imbi12d 227 . . . . 5 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) ↔ (𝑡𝐴𝑡𝑍)))
1413biimpd 136 . . . 4 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) → (𝑡𝐴𝑡𝑍)))
15 eleq1 2116 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝐴𝑢𝐴))
16 eleq1 2116 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝑍𝑢𝑍))
1715, 16imbi12d 227 . . . . 5 (𝑧 = 𝑢 → ((𝑧𝐴𝑧𝑍) ↔ (𝑢𝐴𝑢𝑍)))
1817biimprd 151 . . . 4 (𝑧 = 𝑢 → ((𝑢𝐴𝑢𝑍) → (𝑧𝐴𝑧𝑍)))
196, 7, 8, 9, 10, 14, 18bdsetindis 10453 . . 3 (∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)) → ∀𝑧(𝑧𝐴𝑧𝑍))
201, 19syl6 33 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑧(𝑧𝐴𝑧𝑍)))
21 dfss2 2961 . 2 (𝐴𝑍 ↔ ∀𝑧(𝑧𝐴𝑧𝑍))
2220, 21syl6ibr 155 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639  wal 1257   = wceq 1259  wcel 1409  wral 2323  wrex 2324  wss 2944  c0 3251  suc csuc 4129  BOUNDED wbdc 10319  Ind wind 10409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-bdim 10293  ax-bdsetind 10452
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-suc 4135  df-bdc 10320  df-bj-ind 10410
This theorem is referenced by:  bj-inf2vn  10458
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