Theorem refref 23404

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 2727	. . 3 ⊢ ∪ 𝐴 = ∪ 𝐴
2		ssid 4000	. . . . 5 ⊢ 𝑥 ⊆ 𝑥
3		sseq2 4004	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
4	3	rspcev 3607	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
5	2, 4	mpan2 690	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
6	5	rgen 3058	. . 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 23400	. 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 1534   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065   ⊆ wss 3944  ∪ cuni 4903   class class class wbr 5142  Refcref 23393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-ref 23396

This theorem is referenced by:  locfinref  33378