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Theorem bj-inf2vnlem2 10455
Description: Lemma for bj-inf2vnlem3 10456 and bj-inf2vnlem4 10457. 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 2062 . . . . . . 7 (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
2 eqeq1 2062 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = suc 𝑦𝑢 = suc 𝑦))
32rexbidv 2344 . . . . . . 7 (𝑥 = 𝑢 → (∃𝑦𝐴 𝑥 = suc 𝑦 ↔ ∃𝑦𝐴 𝑢 = suc 𝑦))
41, 3orbi12d 717 . . . . . 6 (𝑥 = 𝑢 → ((𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) ↔ (𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦)))
54rspcv 2669 . . . . 5 (𝑢𝐴 → (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (𝑢 = ∅ ∨ ∃𝑦𝐴 𝑢 = suc 𝑦)))
6 df-bj-ind 10410 . . . . . . . . 9 (Ind 𝑍 ↔ (∅ ∈ 𝑍 ∧ ∀𝑣𝑍 suc 𝑣𝑍))
76simplbi 263 . . . . . . . 8 (Ind 𝑍 → ∅ ∈ 𝑍)
8 eleq1 2116 . . . . . . . 8 (𝑢 = ∅ → (𝑢𝑍 ↔ ∅ ∈ 𝑍))
97, 8syl5ibr 149 . . . . . . 7 (𝑢 = ∅ → (Ind 𝑍𝑢𝑍))
109a1dd 46 . . . . . 6 (𝑢 = ∅ → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
11 vex 2577 . . . . . . . . . 10 𝑦 ∈ V
1211sucid 4181 . . . . . . . . 9 𝑦 ∈ suc 𝑦
13 eleq2 2117 . . . . . . . . . 10 (suc 𝑦 = 𝑢 → (𝑦 ∈ suc 𝑦𝑦𝑢))
1413eqcoms 2059 . . . . . . . . 9 (𝑢 = suc 𝑦 → (𝑦 ∈ suc 𝑦𝑦𝑢))
1512, 14mpbii 140 . . . . . . . 8 (𝑢 = suc 𝑦𝑦𝑢)
16 eleq1 2116 . . . . . . . . . . . . 13 (𝑡 = 𝑦 → (𝑡𝐴𝑦𝐴))
17 eleq1 2116 . . . . . . . . . . . . 13 (𝑡 = 𝑦 → (𝑡𝑍𝑦𝑍))
1816, 17imbi12d 227 . . . . . . . . . . . 12 (𝑡 = 𝑦 → ((𝑡𝐴𝑡𝑍) ↔ (𝑦𝐴𝑦𝑍)))
1918rspcv 2669 . . . . . . . . . . 11 (𝑦𝑢 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝐴𝑦𝑍)))
20 bj-indsuc 10411 . . . . . . . . . . . 12 (Ind 𝑍 → (𝑦𝑍 → suc 𝑦𝑍))
21 eleq1a 2125 . . . . . . . . . . . 12 (suc 𝑦𝑍 → (𝑢 = suc 𝑦𝑢𝑍))
2220, 21syl6com 35 . . . . . . . . . . 11 (𝑦𝑍 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))
2319, 22syl8 69 . . . . . . . . . 10 (𝑦𝑢 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝐴 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))))
2423com13 78 . . . . . . . . 9 (𝑦𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑦𝑢 → (Ind 𝑍 → (𝑢 = suc 𝑦𝑢𝑍)))))
2524com25 89 . . . . . . . 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 2444 . . . . . 6 (∃𝑦𝐴 𝑢 = suc 𝑦 → (Ind 𝑍 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)))
2810, 27jaoi 646 . . . . 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 79 . . 3 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → (𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
3130alrimdv 1772 . 2 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍))))
32 bi2.04 241 . . 3 ((𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)) ↔ (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)))
3332albii 1375 . 2 (∀𝑢(𝑢𝐴 → (∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → 𝑢𝑍)) ↔ ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍)))
3431, 33syl6ib 154 1 (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wo 639  wal 1257   = wceq 1259  wcel 1409  wral 2323  wrex 2324  c0 3251  suc csuc 4129  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
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-sn 3408  df-suc 4135  df-bj-ind 10410
This theorem is referenced by:  bj-inf2vnlem3  10456  bj-inf2vnlem4  10457
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