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Theorem setindtr 43036
Description: Set induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 9774; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
setindtr (∀𝑥(𝑥𝐴𝑥𝐴) → (∃𝑦(Tr 𝑦𝐵𝑦) → 𝐵𝐴))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem setindtr
StepHypRef Expression
1 nfv 1914 . . . . . . . . . . 11 𝑥Tr 𝑦
2 nfa1 2151 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐴𝑥𝐴)
31, 2nfan 1899 . . . . . . . . . 10 𝑥(Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴))
4 eldifn 4132 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑦𝐴) → ¬ 𝑥𝐴)
54adantl 481 . . . . . . . . . . . . 13 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ 𝑥𝐴)
6 trss 5270 . . . . . . . . . . . . . . . . . 18 (Tr 𝑦 → (𝑥𝑦𝑥𝑦))
7 eldifi 4131 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑦𝐴) → 𝑥𝑦)
86, 7impel 505 . . . . . . . . . . . . . . . . 17 ((Tr 𝑦𝑥 ∈ (𝑦𝐴)) → 𝑥𝑦)
9 dfss2 3969 . . . . . . . . . . . . . . . . 17 (𝑥𝑦 ↔ (𝑥𝑦) = 𝑥)
108, 9sylib 218 . . . . . . . . . . . . . . . 16 ((Tr 𝑦𝑥 ∈ (𝑦𝐴)) → (𝑥𝑦) = 𝑥)
1110adantlr 715 . . . . . . . . . . . . . . 15 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → (𝑥𝑦) = 𝑥)
1211sseq1d 4015 . . . . . . . . . . . . . 14 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ((𝑥𝑦) ⊆ 𝐴𝑥𝐴))
13 sp 2183 . . . . . . . . . . . . . . 15 (∀𝑥(𝑥𝐴𝑥𝐴) → (𝑥𝐴𝑥𝐴))
1413ad2antlr 727 . . . . . . . . . . . . . 14 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → (𝑥𝐴𝑥𝐴))
1512, 14sylbid 240 . . . . . . . . . . . . 13 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ((𝑥𝑦) ⊆ 𝐴𝑥𝐴))
165, 15mtod 198 . . . . . . . . . . . 12 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ (𝑥𝑦) ⊆ 𝐴)
17 inssdif0 4374 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝐴 ↔ (𝑥 ∩ (𝑦𝐴)) = ∅)
1816, 17sylnib 328 . . . . . . . . . . 11 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ (𝑥 ∩ (𝑦𝐴)) = ∅)
1918ex 412 . . . . . . . . . 10 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (𝑦𝐴) → ¬ (𝑥 ∩ (𝑦𝐴)) = ∅))
203, 19ralrimi 3257 . . . . . . . . 9 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → ∀𝑥 ∈ (𝑦𝐴) ¬ (𝑥 ∩ (𝑦𝐴)) = ∅)
21 ralnex 3072 . . . . . . . . 9 (∀𝑥 ∈ (𝑦𝐴) ¬ (𝑥 ∩ (𝑦𝐴)) = ∅ ↔ ¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2220, 21sylib 218 . . . . . . . 8 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → ¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
23 vex 3484 . . . . . . . . . . 11 𝑦 ∈ V
2423difexi 5330 . . . . . . . . . 10 (𝑦𝐴) ∈ V
25 zfreg 9635 . . . . . . . . . 10 (((𝑦𝐴) ∈ V ∧ (𝑦𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2624, 25mpan 690 . . . . . . . . 9 ((𝑦𝐴) ≠ ∅ → ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2726necon1bi 2969 . . . . . . . 8 (¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅ → (𝑦𝐴) = ∅)
2822, 27syl 17 . . . . . . 7 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → (𝑦𝐴) = ∅)
29 ssdif0 4366 . . . . . . 7 (𝑦𝐴 ↔ (𝑦𝐴) = ∅)
3028, 29sylibr 234 . . . . . 6 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝑦𝐴)
3130adantlr 715 . . . . 5 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝑦𝐴)
32 simplr 769 . . . . 5 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝐵𝑦)
3331, 32sseldd 3984 . . . 4 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝐵𝐴)
3433ex 412 . . 3 ((Tr 𝑦𝐵𝑦) → (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐵𝐴))
3534exlimiv 1930 . 2 (∃𝑦(Tr 𝑦𝐵𝑦) → (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐵𝐴))
3635com12 32 1 (∀𝑥(𝑥𝐴𝑥𝐴) → (∃𝑦(Tr 𝑦𝐵𝑦) → 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-uni 4908  df-tr 5260
This theorem is referenced by:  setindtrs  43037
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