Theorem refref 22664

Description: Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)

Assertion

Ref	Expression
refref	⊢ (𝐴 ∈ 𝑉 → 𝐴Ref𝐴)

Proof of Theorem refref

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ ∪ 𝐴 = ∪ 𝐴
2		ssid 3943	. . . . 5 ⊢ 𝑥 ⊆ 𝑥
3		sseq2 3947	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
4	3	rspcev 3561	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
5	2, 4	mpan2 688	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
6	5	rgen 3074	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦
7	1, 6	pm3.2i 471	. 2 ⊢ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
8	1, 1	isref 22660	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐴 ↔ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
9	7, 8	mpbiri 257	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Ref𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  Refcref 22653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-ref 22656

This theorem is referenced by:  locfinref  31791