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Theorem bj-inf2vnlem2 11209
Description: Lemma for bj-inf2vnlem3 11210 and bj-inf2vnlem4 11211. 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 2089 . . . . . . 7 (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
2 eqeq1 2089 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = suc 𝑦𝑢 = suc 𝑦))
32rexbidv 2375 . . . . . . 7 (𝑥 = 𝑢 → (∃𝑦𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦𝐴 𝑢 = suc 𝑦))
41, 3orbi12d 740 . . . . . 6 (𝑥 = 𝑢 → ((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) ↔ (𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦)))
54rspcv 2708 . . . . 5 (𝑢𝐴 → (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦)))
6 df-bj-ind 11165 . . . . . . . . 9 (Ind 𝑍 ↔ (∅ ∈ 𝑍 ∧ ∀𝑣𝑍 suc 𝑣𝑍))
76simplbi 268 . . . . . . . 8 (Ind 𝑍 → ∅ ∈ 𝑍)
8 eleq1 2145 . . . . . . . 8 (𝑢 = ∅ → (𝑢𝑍 ↔ ∅ ∈ 𝑍))
97, 8syl5ibr 154 . . . . . . 7 (𝑢 = ∅ → (Ind 𝑍𝑢𝑍))
109a1dd 47 . . . . . 6 (𝑢 = ∅ → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
11 vex 2615 . . . . . . . . . 10 𝑦 ∈ V
1211sucid 4208 . . . . . . . . 9 𝑦 ∈ suc 𝑦
13 eleq2 2146 . . . . . . . . . 10 (suc 𝑦 = 𝑢 → (𝑦 ∈ suc 𝑦𝑦𝑢))
1413eqcoms 2086 . . . . . . . . 9 (𝑢 = suc 𝑦 → (𝑦 ∈ suc 𝑦𝑦𝑢))
1512, 14mpbii 146 . . . . . . . 8 (𝑢 = suc 𝑦𝑦𝑢)
16 eleq1 2145 . . . . . . . . . . . . 13 (𝑡 = 𝑦 → (𝑡𝐴𝑦𝐴))
17 eleq1 2145 . . . . . . . . . . . . 13 (𝑡 = 𝑦 → (𝑡𝑍𝑦𝑍))
1816, 17imbi12d 232 . . . . . . . . . . . 12 (𝑡 = 𝑦 → ((𝑡𝐴𝑡𝑍) ↔ (𝑦𝐴𝑦𝑍)))
1918rspcv 2708 . . . . . . . . . . 11 (𝑦𝑢 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝐴𝑦𝑍)))
20 bj-indsuc 11166 . . . . . . . . . . . 12 (Ind 𝑍 → (𝑦𝑍 → suc 𝑦𝑍))
21 eleq1a 2154 . . . . . . . . . . . 12 (suc 𝑦𝑍 → (𝑢 = suc 𝑦𝑢𝑍))
2220, 21syl6com 35 . . . . . . . . . . 11 (𝑦𝑍 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))
2319, 22syl8 70 . . . . . . . . . 10 (𝑦𝑢 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝐴 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))))
2423com13 79 . . . . . . . . 9 (𝑦𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝑢 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))))
2524com25 90 . . . . . . . 8 (𝑦𝐴 → (𝑢 = suc 𝑦 → (𝑦𝑢 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))))
2615, 25mpdi 42 . . . . . . 7 (𝑦𝐴 → (𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
2726rexlimiv 2477 . . . . . 6 (∃𝑦𝐴 𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
2810, 27jaoi 669 . . . . 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 80 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → (𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
3130alrimdv 1799 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
32 bi2.04 246 . . 3 ((𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)) ↔ (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)))
3332albii 1400 . 2 (∀𝑢(𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)) ↔ ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)))
3431, 33syl6ib 159 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 662  wal 1283   = wceq 1285  wcel 1434  wral 2353  wrex 2354  c0 3269  suc csuc 4156  Ind wind 11164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-sn 3428  df-suc 4162  df-bj-ind 11165
This theorem is referenced by:  bj-inf2vnlem3  11210  bj-inf2vnlem4  11211
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