Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-inf2vnlem4 GIF version

Theorem bj-inf2vnlem4 15108
Description: Lemma for bj-inf2vn2 15110. (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 15106 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
2 nfv 1538 . . . 4 𝑧(𝑡𝐴𝑡𝑍)
3 nfv 1538 . . . 4 𝑧(𝑢𝐴𝑢𝑍)
4 nfv 1538 . . . 4 𝑢(𝑧𝐴𝑧𝑍)
5 nfv 1538 . . . 4 𝑢(𝑡𝐴𝑡𝑍)
6 eleq1 2251 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝐴𝑡𝐴))
7 eleq1 2251 . . . . . 6 (𝑧 = 𝑡 → (𝑧𝑍𝑡𝑍))
86, 7imbi12d 234 . . . . 5 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) ↔ (𝑡𝐴𝑡𝑍)))
98biimpd 144 . . . 4 (𝑧 = 𝑡 → ((𝑧𝐴𝑧𝑍) → (𝑡𝐴𝑡𝑍)))
10 eleq1 2251 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝐴𝑢𝐴))
11 eleq1 2251 . . . . . 6 (𝑧 = 𝑢 → (𝑧𝑍𝑢𝑍))
1210, 11imbi12d 234 . . . . 5 (𝑧 = 𝑢 → ((𝑧𝐴𝑧𝑍) ↔ (𝑢𝐴𝑢𝑍)))
1312biimprd 158 . . . 4 (𝑧 = 𝑢 → ((𝑢𝐴𝑢𝑍) → (𝑧𝐴𝑧𝑍)))
142, 3, 4, 5, 9, 13setindis 15102 . . 3 (∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)) → ∀𝑧(𝑧𝐴𝑧𝑍))
151, 14syl6 33 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑧(𝑧𝐴𝑧𝑍)))
16 dfss2 3158 . 2 (𝐴𝑍 ↔ ∀𝑧(𝑧𝐴𝑧𝑍))
1715, 16imbitrrdi 162 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 709  wal 1361   = wceq 1363  wcel 2159  wral 2467  wrex 2468  wss 3143  c0 3436  suc csuc 4379  Ind wind 15061
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170  ax-setind 4550
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-sn 3612  df-suc 4385  df-bj-ind 15062
This theorem is referenced by:  bj-inf2vn2  15110
  Copyright terms: Public domain W3C validator