Theorem trficl 44713

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 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧}

Assertion

Ref	Expression
trficl	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴

Allowed substitution hints:   𝐴(𝑥, 𝑧)

Proof of Theorem trficl

Step	Hyp	Ref	Expression
1		trficl.a	. 2 ⊢ 𝐴 = {𝑧 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧}
2		vex 3454	. . 3 ⊢ 𝑥 ∈ V
3	2	inex1 5276	. 2 ⊢ (𝑥 ∩ 𝑦) ∈ V
4		id 23	. . . 4 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → 𝑧 = (𝑥 ∩ 𝑦))
5	4, 4	coeq12d 5841	. . 3 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ∘ 𝑧) = ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)))
6	5, 4	sseq12d 3941	. 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)) ⊆ (𝑥 ∩ 𝑦)))
7		id 23	. . . 4 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
8	7, 7	coeq12d 5841	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∘ 𝑧) = (𝑥 ∘ 𝑥))
9	8, 7	sseq12d 3941	. 2 ⊢ (𝑧 = 𝑥 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑥 ∘ 𝑥) ⊆ 𝑥))
10		id 23	. . . 4 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
11	10, 10	coeq12d 5841	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∘ 𝑧) = (𝑦 ∘ 𝑦))
12	11, 10	sseq12d 3941	. 2 ⊢ (𝑧 = 𝑦 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑦 ∘ 𝑦) ⊆ 𝑦))
13		trin2 6116	. 2 ⊢ (((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦) → ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)) ⊆ (𝑥 ∩ 𝑦))
14	1, 3, 6, 9, 12, 13	cllem0 44611	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∘ ccom 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-co 5659

This theorem is used by: (None)