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Theorem setindtr 39499
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 9164; 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 1906 . . . . . . . . . . 11 𝑥Tr 𝑦
2 nfa1 2146 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐴𝑥𝐴)
31, 2nfan 1891 . . . . . . . . . 10 𝑥(Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴))
4 eldifn 4101 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑦𝐴) → ¬ 𝑥𝐴)
54adantl 482 . . . . . . . . . . . . 13 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ 𝑥𝐴)
6 trss 5172 . . . . . . . . . . . . . . . . . 18 (Tr 𝑦 → (𝑥𝑦𝑥𝑦))
7 eldifi 4100 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑦𝐴) → 𝑥𝑦)
86, 7impel 506 . . . . . . . . . . . . . . . . 17 ((Tr 𝑦𝑥 ∈ (𝑦𝐴)) → 𝑥𝑦)
9 df-ss 3949 . . . . . . . . . . . . . . . . 17 (𝑥𝑦 ↔ (𝑥𝑦) = 𝑥)
108, 9sylib 219 . . . . . . . . . . . . . . . 16 ((Tr 𝑦𝑥 ∈ (𝑦𝐴)) → (𝑥𝑦) = 𝑥)
1110adantlr 711 . . . . . . . . . . . . . . 15 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → (𝑥𝑦) = 𝑥)
1211sseq1d 3995 . . . . . . . . . . . . . 14 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ((𝑥𝑦) ⊆ 𝐴𝑥𝐴))
13 sp 2172 . . . . . . . . . . . . . . 15 (∀𝑥(𝑥𝐴𝑥𝐴) → (𝑥𝐴𝑥𝐴))
1413ad2antlr 723 . . . . . . . . . . . . . 14 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → (𝑥𝐴𝑥𝐴))
1512, 14sylbid 241 . . . . . . . . . . . . 13 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ((𝑥𝑦) ⊆ 𝐴𝑥𝐴))
165, 15mtod 199 . . . . . . . . . . . 12 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ (𝑥𝑦) ⊆ 𝐴)
17 inssdif0 4326 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝐴 ↔ (𝑥 ∩ (𝑦𝐴)) = ∅)
1816, 17sylnib 329 . . . . . . . . . . 11 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ (𝑥 ∩ (𝑦𝐴)) = ∅)
1918ex 413 . . . . . . . . . 10 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (𝑦𝐴) → ¬ (𝑥 ∩ (𝑦𝐴)) = ∅))
203, 19ralrimi 3213 . . . . . . . . 9 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → ∀𝑥 ∈ (𝑦𝐴) ¬ (𝑥 ∩ (𝑦𝐴)) = ∅)
21 ralnex 3233 . . . . . . . . 9 (∀𝑥 ∈ (𝑦𝐴) ¬ (𝑥 ∩ (𝑦𝐴)) = ∅ ↔ ¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2220, 21sylib 219 . . . . . . . 8 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → ¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
23 vex 3495 . . . . . . . . . . 11 𝑦 ∈ V
2423difexi 5223 . . . . . . . . . 10 (𝑦𝐴) ∈ V
25 zfreg 9047 . . . . . . . . . 10 (((𝑦𝐴) ∈ V ∧ (𝑦𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2624, 25mpan 686 . . . . . . . . 9 ((𝑦𝐴) ≠ ∅ → ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2726necon1bi 3041 . . . . . . . 8 (¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅ → (𝑦𝐴) = ∅)
2822, 27syl 17 . . . . . . 7 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → (𝑦𝐴) = ∅)
29 ssdif0 4320 . . . . . . 7 (𝑦𝐴 ↔ (𝑦𝐴) = ∅)
3028, 29sylibr 235 . . . . . 6 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝑦𝐴)
3130adantlr 711 . . . . 5 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝑦𝐴)
32 simplr 765 . . . . 5 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝐵𝑦)
3331, 32sseldd 3965 . . . 4 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝐵𝐴)
3433ex 413 . . 3 ((Tr 𝑦𝐵𝑦) → (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐵𝐴))
3534exlimiv 1922 . 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 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  cin 3932  wss 3933  c0 4288  Tr wtr 5163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-uni 4831  df-tr 5164
This theorem is referenced by:  setindtrs  39500
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