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 2756	. . 3 ⊢ ∪ 𝐴 = ∪ 𝐴
2		ssid 3953	. . . . 5 ⊢ 𝑥 ⊆ 𝑥
3		sseq2 3957	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥))
4	3	rspcev 3576	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
5	2, 4	mpan2 699	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
6	5	rgen 3072	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦
7	1, 6	pm3.2i 473	. 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 260	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Ref𝐴)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∀wral 3070  ∃wrex 3080   ⊆ wss 3899  ∪ cuni 4859   class class class wbr 5094  Refcref 23535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-xp 5646  df-rel 5647  df-ref 23538

This theorem is referenced by:  locfinref  34092