Theorem refref 23546

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 2737	. . 3 ⊢ ∪ 𝐴 = ∪ 𝐴
2		ssid 4021	. . . . 5 ⊢ 𝑥 ⊆ 𝑥
3		sseq2 4025	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
4	3	rspcev 3625	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
5	2, 4	mpan2 691	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
6	5	rgen 3063	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦
7	1, 6	pm3.2i 470	. 2 ⊢ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
8	1, 1	isref 23542	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐴 ↔ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)))
9	7, 8	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Ref𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3966  ∪ cuni 4915   class class class wbr 5151  Refcref 23535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-xp 5699  df-rel 5700  df-ref 23538

This theorem is referenced by:  locfinref  33834