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Theorem trficl 40743
 Description: The class of all transitive relations has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 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 3413 . . 3 𝑥 ∈ V
32inex1 5187 . 2 (𝑥𝑦) ∈ V
4 id 22 . . . 4 (𝑧 = (𝑥𝑦) → 𝑧 = (𝑥𝑦))
54, 4coeq12d 5704 . . 3 (𝑧 = (𝑥𝑦) → (𝑧𝑧) = ((𝑥𝑦) ∘ (𝑥𝑦)))
65, 4sseq12d 3925 . 2 (𝑧 = (𝑥𝑦) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((𝑥𝑦) ∘ (𝑥𝑦)) ⊆ (𝑥𝑦)))
7 id 22 . . . 4 (𝑧 = 𝑥𝑧 = 𝑥)
87, 7coeq12d 5704 . . 3 (𝑧 = 𝑥 → (𝑧𝑧) = (𝑥𝑥))
98, 7sseq12d 3925 . 2 (𝑧 = 𝑥 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑥𝑥) ⊆ 𝑥))
10 id 22 . . . 4 (𝑧 = 𝑦𝑧 = 𝑦)
1110, 10coeq12d 5704 . . 3 (𝑧 = 𝑦 → (𝑧𝑧) = (𝑦𝑦))
1211, 10sseq12d 3925 . 2 (𝑧 = 𝑦 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑦𝑦) ⊆ 𝑦))
13 trin2 5955 . 2 (((𝑥𝑥) ⊆ 𝑥 ∧ (𝑦𝑦) ⊆ 𝑦) → ((𝑥𝑦) ∘ (𝑥𝑦)) ⊆ (𝑥𝑦))
141, 3, 6, 9, 12, 13cllem0 40638 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858   ∘ ccom 5528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-rel 5531  df-co 5533 This theorem is referenced by: (None)
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