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Theorem trficl 37439
Description: The class of all transitive relations has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
Hypothesis
Ref Expression
trficl.a 𝐴 = {𝑧 ∣ (𝑧𝑧) ⊆ 𝑧}
Assertion
Ref Expression
trficl 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴
Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem trficl
StepHypRef Expression
1 trficl.a . 2 𝐴 = {𝑧 ∣ (𝑧𝑧) ⊆ 𝑧}
2 vex 3189 . . 3 𝑥 ∈ V
32inex1 4759 . 2 (𝑥𝑦) ∈ V
4 id 22 . . . 4 (𝑧 = (𝑥𝑦) → 𝑧 = (𝑥𝑦))
54, 4coeq12d 5246 . . 3 (𝑧 = (𝑥𝑦) → (𝑧𝑧) = ((𝑥𝑦) ∘ (𝑥𝑦)))
65, 4sseq12d 3613 . 2 (𝑧 = (𝑥𝑦) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((𝑥𝑦) ∘ (𝑥𝑦)) ⊆ (𝑥𝑦)))
7 id 22 . . . 4 (𝑧 = 𝑥𝑧 = 𝑥)
87, 7coeq12d 5246 . . 3 (𝑧 = 𝑥 → (𝑧𝑧) = (𝑥𝑥))
98, 7sseq12d 3613 . 2 (𝑧 = 𝑥 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑥𝑥) ⊆ 𝑥))
10 id 22 . . . 4 (𝑧 = 𝑦𝑧 = 𝑦)
1110, 10coeq12d 5246 . . 3 (𝑧 = 𝑦 → (𝑧𝑧) = (𝑦𝑦))
1211, 10sseq12d 3613 . 2 (𝑧 = 𝑦 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑦𝑦) ⊆ 𝑦))
13 trin2 5478 . 2 (((𝑥𝑥) ⊆ 𝑥 ∧ (𝑦𝑦) ⊆ 𝑦) → ((𝑥𝑦) ∘ (𝑥𝑦)) ⊆ (𝑥𝑦))
141, 3, 6, 9, 12, 13cllem0 37349 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3186  cin 3554  wss 3555  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-co 5083
This theorem is referenced by: (None)
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