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Theorem trficl 36980
 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 3176 . . 3 𝑥 ∈ V
32inex1 4727 . 2 (𝑥𝑦) ∈ V
4 id 22 . . . 4 (𝑧 = (𝑥𝑦) → 𝑧 = (𝑥𝑦))
54, 4coeq12d 5208 . . 3 (𝑧 = (𝑥𝑦) → (𝑧𝑧) = ((𝑥𝑦) ∘ (𝑥𝑦)))
65, 4sseq12d 3597 . 2 (𝑧 = (𝑥𝑦) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((𝑥𝑦) ∘ (𝑥𝑦)) ⊆ (𝑥𝑦)))
7 id 22 . . . 4 (𝑧 = 𝑥𝑧 = 𝑥)
87, 7coeq12d 5208 . . 3 (𝑧 = 𝑥 → (𝑧𝑧) = (𝑥𝑥))
98, 7sseq12d 3597 . 2 (𝑧 = 𝑥 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑥𝑥) ⊆ 𝑥))
10 id 22 . . . 4 (𝑧 = 𝑦𝑧 = 𝑦)
1110, 10coeq12d 5208 . . 3 (𝑧 = 𝑦 → (𝑧𝑧) = (𝑦𝑦))
1211, 10sseq12d 3597 . 2 (𝑧 = 𝑦 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑦𝑦) ⊆ 𝑦))
13 trin2 5438 . 2 (((𝑥𝑥) ⊆ 𝑥 ∧ (𝑦𝑦) ⊆ 𝑦) → ((𝑥𝑦) ∘ (𝑥𝑦)) ⊆ (𝑥𝑦))
141, 3, 6, 9, 12, 13cllem0 36890 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-co 5047 This theorem is referenced by: (None)
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