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Theorem bj-inf2vnlem2 13065
Description: Lemma for bj-inf2vnlem3 13066 and bj-inf2vnlem4 13067. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inf2vnlem2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
Distinct variable groups:   𝑥,𝑦,𝑡,𝑢,𝐴   𝑥,𝑍,𝑦,𝑡,𝑢

Proof of Theorem bj-inf2vnlem2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2124 . . . . . . 7 (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
2 eqeq1 2124 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = suc 𝑦𝑢 = suc 𝑦))
32rexbidv 2415 . . . . . . 7 (𝑥 = 𝑢 → (∃𝑦𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦𝐴 𝑢 = suc 𝑦))
41, 3orbi12d 767 . . . . . 6 (𝑥 = 𝑢 → ((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) ↔ (𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦)))
54rspcv 2759 . . . . 5 (𝑢𝐴 → (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦)))
6 df-bj-ind 13021 . . . . . . . . 9 (Ind 𝑍 ↔ (∅ ∈ 𝑍 ∧ ∀𝑣𝑍 suc 𝑣𝑍))
76simplbi 272 . . . . . . . 8 (Ind 𝑍 → ∅ ∈ 𝑍)
8 eleq1 2180 . . . . . . . 8 (𝑢 = ∅ → (𝑢𝑍 ↔ ∅ ∈ 𝑍))
97, 8syl5ibr 155 . . . . . . 7 (𝑢 = ∅ → (Ind 𝑍𝑢𝑍))
109a1dd 48 . . . . . 6 (𝑢 = ∅ → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
11 vex 2663 . . . . . . . . . 10 𝑦 ∈ V
1211sucid 4309 . . . . . . . . 9 𝑦 ∈ suc 𝑦
13 eleq2 2181 . . . . . . . . . 10 (suc 𝑦 = 𝑢 → (𝑦 ∈ suc 𝑦𝑦𝑢))
1413eqcoms 2120 . . . . . . . . 9 (𝑢 = suc 𝑦 → (𝑦 ∈ suc 𝑦𝑦𝑢))
1512, 14mpbii 147 . . . . . . . 8 (𝑢 = suc 𝑦𝑦𝑢)
16 eleq1 2180 . . . . . . . . . . . . 13 (𝑡 = 𝑦 → (𝑡𝐴𝑦𝐴))
17 eleq1 2180 . . . . . . . . . . . . 13 (𝑡 = 𝑦 → (𝑡𝑍𝑦𝑍))
1816, 17imbi12d 233 . . . . . . . . . . . 12 (𝑡 = 𝑦 → ((𝑡𝐴𝑡𝑍) ↔ (𝑦𝐴𝑦𝑍)))
1918rspcv 2759 . . . . . . . . . . 11 (𝑦𝑢 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝐴𝑦𝑍)))
20 bj-indsuc 13022 . . . . . . . . . . . 12 (Ind 𝑍 → (𝑦𝑍 → suc 𝑦𝑍))
21 eleq1a 2189 . . . . . . . . . . . 12 (suc 𝑦𝑍 → (𝑢 = suc 𝑦𝑢𝑍))
2220, 21syl6com 35 . . . . . . . . . . 11 (𝑦𝑍 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))
2319, 22syl8 71 . . . . . . . . . 10 (𝑦𝑢 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝐴 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))))
2423com13 80 . . . . . . . . 9 (𝑦𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝑢 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))))
2524com25 91 . . . . . . . 8 (𝑦𝐴 → (𝑢 = suc 𝑦 → (𝑦𝑢 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))))
2615, 25mpdi 43 . . . . . . 7 (𝑦𝐴 → (𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
2726rexlimiv 2520 . . . . . 6 (∃𝑦𝐴 𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
2810, 27jaoi 690 . . . . 5 ((𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦) → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
295, 28syl6 33 . . . 4 (𝑢𝐴 → (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
3029com3l 81 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → (𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
3130alrimdv 1832 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
32 bi2.04 247 . . 3 ((𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)) ↔ (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)))
3332albii 1431 . 2 (∀𝑢(𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)) ↔ ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)))
3431, 33syl6ib 160 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 682  wal 1314   = wceq 1316  wcel 1465  wral 2393  wrex 2394  c0 3333  suc csuc 4257  Ind wind 13020
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-suc 4263  df-bj-ind 13021
This theorem is referenced by:  bj-inf2vnlem3  13066  bj-inf2vnlem4  13067
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