Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trficl Structured version   Visualization version   GIF version

Theorem trficl 43241
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 3465 . . 3 𝑥 ∈ V
32inex1 5318 . 2 (𝑥𝑦) ∈ V
4 id 22 . . . 4 (𝑧 = (𝑥𝑦) → 𝑧 = (𝑥𝑦))
54, 4coeq12d 5867 . . 3 (𝑧 = (𝑥𝑦) → (𝑧𝑧) = ((𝑥𝑦) ∘ (𝑥𝑦)))
65, 4sseq12d 4010 . 2 (𝑧 = (𝑥𝑦) → ((𝑧𝑧) ⊆ 𝑧 ↔ ((𝑥𝑦) ∘ (𝑥𝑦)) ⊆ (𝑥𝑦)))
7 id 22 . . . 4 (𝑧 = 𝑥𝑧 = 𝑥)
87, 7coeq12d 5867 . . 3 (𝑧 = 𝑥 → (𝑧𝑧) = (𝑥𝑥))
98, 7sseq12d 4010 . 2 (𝑧 = 𝑥 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑥𝑥) ⊆ 𝑥))
10 id 22 . . . 4 (𝑧 = 𝑦𝑧 = 𝑦)
1110, 10coeq12d 5867 . . 3 (𝑧 = 𝑦 → (𝑧𝑧) = (𝑦𝑦))
1211, 10sseq12d 4010 . 2 (𝑧 = 𝑦 → ((𝑧𝑧) ⊆ 𝑧 ↔ (𝑦𝑦) ⊆ 𝑦))
13 trin2 6130 . 2 (((𝑥𝑥) ⊆ 𝑥 ∧ (𝑦𝑦) ⊆ 𝑦) → ((𝑥𝑦) ∘ (𝑥𝑦)) ⊆ (𝑥𝑦))
141, 3, 6, 9, 12, 13cllem0 43138 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2702  wral 3050  Vcvv 3461  cin 3943  wss 3944  ccom 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-co 5687
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator