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Theorem bj-inf2vnlem2 10096
 Description: Lemma for bj-inf2vnlem3 10097 and bj-inf2vnlem4 10098. 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 2046 . . . . . . 7 (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
2 eqeq1 2046 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = suc 𝑦𝑢 = suc 𝑦))
32rexbidv 2327 . . . . . . 7 (𝑥 = 𝑢 → (∃𝑦𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦𝐴 𝑢 = suc 𝑦))
41, 3orbi12d 707 . . . . . 6 (𝑥 = 𝑢 → ((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) ↔ (𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦)))
54rspcv 2652 . . . . 5 (𝑢𝐴 → (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦)))
6 df-bj-ind 10051 . . . . . . . . 9 (Ind 𝑍 ↔ (∅ ∈ 𝑍 ∧ ∀𝑣𝑍 suc 𝑣𝑍))
76simplbi 259 . . . . . . . 8 (Ind 𝑍 → ∅ ∈ 𝑍)
8 eleq1 2100 . . . . . . . 8 (𝑢 = ∅ → (𝑢𝑍 ↔ ∅ ∈ 𝑍))
97, 8syl5ibr 145 . . . . . . 7 (𝑢 = ∅ → (Ind 𝑍𝑢𝑍))
109a1dd 42 . . . . . 6 (𝑢 = ∅ → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
11 vex 2560 . . . . . . . . . 10 𝑦 ∈ V
1211sucid 4154 . . . . . . . . 9 𝑦 ∈ suc 𝑦
13 eleq2 2101 . . . . . . . . . 10 (suc 𝑦 = 𝑢 → (𝑦 ∈ suc 𝑦𝑦𝑢))
1413eqcoms 2043 . . . . . . . . 9 (𝑢 = suc 𝑦 → (𝑦 ∈ suc 𝑦𝑦𝑢))
1512, 14mpbii 136 . . . . . . . 8 (𝑢 = suc 𝑦𝑦𝑢)
16 eleq1 2100 . . . . . . . . . . . . 13 (𝑡 = 𝑦 → (𝑡𝐴𝑦𝐴))
17 eleq1 2100 . . . . . . . . . . . . 13 (𝑡 = 𝑦 → (𝑡𝑍𝑦𝑍))
1816, 17imbi12d 223 . . . . . . . . . . . 12 (𝑡 = 𝑦 → ((𝑡𝐴𝑡𝑍) ↔ (𝑦𝐴𝑦𝑍)))
1918rspcv 2652 . . . . . . . . . . 11 (𝑦𝑢 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝐴𝑦𝑍)))
20 bj-indsuc 10052 . . . . . . . . . . . 12 (Ind 𝑍 → (𝑦𝑍 → suc 𝑦𝑍))
21 eleq1a 2109 . . . . . . . . . . . 12 (suc 𝑦𝑍 → (𝑢 = suc 𝑦𝑢𝑍))
2220, 21syl6com 31 . . . . . . . . . . 11 (𝑦𝑍 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))
2319, 22syl8 65 . . . . . . . . . 10 (𝑦𝑢 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝐴 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))))
2423com13 74 . . . . . . . . 9 (𝑦𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝑢 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))))
2524com25 85 . . . . . . . 8 (𝑦𝐴 → (𝑢 = suc 𝑦 → (𝑦𝑢 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))))
2615, 25mpdi 38 . . . . . . 7 (𝑦𝐴 → (𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
2726rexlimiv 2427 . . . . . 6 (∃𝑦𝐴 𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
2810, 27jaoi 636 . . . . 5 ((𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦) → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
295, 28syl6 29 . . . 4 (𝑢𝐴 → (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
3029com3l 75 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → (𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
3130alrimdv 1756 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
32 bi2.04 237 . . 3 ((𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)) ↔ (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)))
3332albii 1359 . 2 (∀𝑢(𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)) ↔ ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)))
3431, 33syl6ib 150 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 629  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  ∅c0 3224  suc csuc 4102  Ind wind 10050 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-suc 4108  df-bj-ind 10051 This theorem is referenced by:  bj-inf2vnlem3  10097  bj-inf2vnlem4  10098
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