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Theorem setindtr 41097
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 9583; 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 1916 . . . . . . . . . . 11 𝑥Tr 𝑦
2 nfa1 2147 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐴𝑥𝐴)
31, 2nfan 1901 . . . . . . . . . 10 𝑥(Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴))
4 eldifn 4073 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑦𝐴) → ¬ 𝑥𝐴)
54adantl 482 . . . . . . . . . . . . 13 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ 𝑥𝐴)
6 trss 5217 . . . . . . . . . . . . . . . . . 18 (Tr 𝑦 → (𝑥𝑦𝑥𝑦))
7 eldifi 4072 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑦𝐴) → 𝑥𝑦)
86, 7impel 506 . . . . . . . . . . . . . . . . 17 ((Tr 𝑦𝑥 ∈ (𝑦𝐴)) → 𝑥𝑦)
9 df-ss 3914 . . . . . . . . . . . . . . . . 17 (𝑥𝑦 ↔ (𝑥𝑦) = 𝑥)
108, 9sylib 217 . . . . . . . . . . . . . . . 16 ((Tr 𝑦𝑥 ∈ (𝑦𝐴)) → (𝑥𝑦) = 𝑥)
1110adantlr 712 . . . . . . . . . . . . . . 15 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → (𝑥𝑦) = 𝑥)
1211sseq1d 3962 . . . . . . . . . . . . . 14 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ((𝑥𝑦) ⊆ 𝐴𝑥𝐴))
13 sp 2175 . . . . . . . . . . . . . . 15 (∀𝑥(𝑥𝐴𝑥𝐴) → (𝑥𝐴𝑥𝐴))
1413ad2antlr 724 . . . . . . . . . . . . . 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 4315 . . . . . . . . . . . 12 ((𝑥𝑦) ⊆ 𝐴 ↔ (𝑥 ∩ (𝑦𝐴)) = ∅)
1816, 17sylnib 327 . . . . . . . . . . 11 (((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (𝑦𝐴)) → ¬ (𝑥 ∩ (𝑦𝐴)) = ∅)
1918ex 413 . . . . . . . . . 10 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (𝑦𝐴) → ¬ (𝑥 ∩ (𝑦𝐴)) = ∅))
203, 19ralrimi 3236 . . . . . . . . 9 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → ∀𝑥 ∈ (𝑦𝐴) ¬ (𝑥 ∩ (𝑦𝐴)) = ∅)
21 ralnex 3072 . . . . . . . . 9 (∀𝑥 ∈ (𝑦𝐴) ¬ (𝑥 ∩ (𝑦𝐴)) = ∅ ↔ ¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2220, 21sylib 217 . . . . . . . 8 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → ¬ ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
23 vex 3445 . . . . . . . . . . 11 𝑦 ∈ V
2423difexi 5269 . . . . . . . . . 10 (𝑦𝐴) ∈ V
25 zfreg 9444 . . . . . . . . . 10 (((𝑦𝐴) ∈ V ∧ (𝑦𝐴) ≠ ∅) → ∃𝑥 ∈ (𝑦𝐴)(𝑥 ∩ (𝑦𝐴)) = ∅)
2624, 25mpan 687 . . . . . . . . 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 4309 . . . . . . 7 (𝑦𝐴 ↔ (𝑦𝐴) = ∅)
3028, 29sylibr 233 . . . . . 6 ((Tr 𝑦 ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝑦𝐴)
3130adantlr 712 . . . . 5 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝑦𝐴)
32 simplr 766 . . . . 5 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝐵𝑦)
3331, 32sseldd 3932 . . . 4 (((Tr 𝑦𝐵𝑦) ∧ ∀𝑥(𝑥𝐴𝑥𝐴)) → 𝐵𝐴)
3433ex 413 . . 3 ((Tr 𝑦𝐵𝑦) → (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐵𝐴))
3534exlimiv 1932 . 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 1538   = wceq 1540  wex 1780  wcel 2105  wne 2940  wral 3061  wrex 3070  Vcvv 3441  cdif 3894  cin 3896  wss 3897  c0 4268  Tr wtr 5206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-reg 9441
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-in 3904  df-ss 3914  df-nul 4269  df-uni 4852  df-tr 5207
This theorem is referenced by:  setindtrs  41098
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