Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  setindtr Structured version   Visualization version   GIF version

Theorem setindtr 42682
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 9777; 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 1910 . . . . . . . . . . 11 𝑥Tr 𝑦
2 nfa1 2141 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐴𝑥𝐴)
31, 2nfan 1895 . . . . . . . . . 10 𝑥(Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴))
4 eldifn 4127 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑦𝐴) → ¬ 𝑥𝐴)
54adantl 480 . . . . . . . . . . . . 13 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ 𝑥𝐴)
6 trss 5281 . . . . . . . . . . . . . . . . . 18 (Tr 𝑦 → (𝑥𝑦𝑥𝑦))
7 eldifi 4126 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑦𝐴) → 𝑥𝑦)
86, 7impel 504 . . . . . . . . . . . . . . . . 17 ((Tr 𝑦𝑥 ∈ (𝑦𝐴)) → 𝑥𝑦)
9 dfss2 3965 . . . . . . . . . . . . . . . . 17 (𝑥𝑦 ↔ (𝑥𝑦) = 𝑥)
108, 9sylib 217 . . . . . . . . . . . . . . . 16 ((Tr 𝑦𝑥 ∈ (𝑦𝐴)) → (𝑥𝑦) = 𝑥)
1110adantlr 713 . . . . . . . . . . . . . . 15 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → (𝑥𝑦) = 𝑥)
1211sseq1d 4011 . . . . . . . . . . . . . 14 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ((𝑥𝑦) ⊆ 𝐴𝑥𝐴))
13 sp 2172 . . . . . . . . . . . . . . 15 (∀𝑥(𝑥𝐴𝑥𝐴) → (𝑥𝐴𝑥𝐴))
1413ad2antlr 725 . . . . . . . . . . . . . 14 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → (𝑥𝐴𝑥𝐴))
1512, 14sylbid 239 . . . . . . . . . . . . 13 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ((𝑥𝑦) ⊆ 𝐴𝑥𝐴))
165, 15mtod 197 . . . . . . . . . . . 12 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ (𝑥𝑦) ⊆ 𝐴)
17 inssdif0 4374 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝐴 ↔ (𝑥 ∩ (𝑦𝐴)) = ∅)
1816, 17sylnib 327 . . . . . . . . . . 11 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ (𝑥 ∩ (𝑦𝐴)) = ∅)
1918ex 411 . . . . . . . . . 10 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (𝑦𝐴) → ¬ (𝑥 ∩ (𝑦𝐴)) = ∅))
203, 19ralrimi 3245 . . . . . . . . 9 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → ∀𝑥 ∈ (𝑦𝐴) ¬ (𝑥 ∩ (𝑦𝐴)) = ∅)
21 ralnex 3062 . . . . . . . . 9 (∀𝑥 ∈ (𝑦𝐴) ¬ (𝑥 ∩ (𝑦𝐴)) = ∅ ↔ ¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2220, 21sylib 217 . . . . . . . 8 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → ¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
23 vex 3466 . . . . . . . . . . 11 𝑦 ∈ V
2423difexi 5335 . . . . . . . . . 10 (𝑦𝐴) ∈ V
25 zfreg 9638 . . . . . . . . . 10 (((𝑦𝐴) ∈ V ∧ (𝑦𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2624, 25mpan 688 . . . . . . . . 9 ((𝑦𝐴) ≠ ∅ → ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2726necon1bi 2959 . . . . . . . 8 (¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅ → (𝑦𝐴) = ∅)
2822, 27syl 17 . . . . . . 7 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → (𝑦𝐴) = ∅)
29 ssdif0 4366 . . . . . . 7 (𝑦𝐴 ↔ (𝑦𝐴) = ∅)
3028, 29sylibr 233 . . . . . 6 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝑦𝐴)
3130adantlr 713 . . . . 5 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝑦𝐴)
32 simplr 767 . . . . 5 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝐵𝑦)
3331, 32sseldd 3980 . . . 4 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝐵𝐴)
3433ex 411 . . 3 ((Tr 𝑦𝐵𝑦) → (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐵𝐴))
3534exlimiv 1926 . 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 394  wal 1532   = wceq 1534  wex 1774  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3462  cdif 3944  cin 3946  wss 3947  c0 4325  Tr wtr 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-reg 9635
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4326  df-uni 4914  df-tr 5271
This theorem is referenced by:  setindtrs  42683
  Copyright terms: Public domain W3C validator