Theorem refun0 22666

Description: Adding the empty set preserves refinements. (Contributed by Thierry Arnoux, 31-Jan-2020.)

Assertion

Ref	Expression
refun0	⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Proof of Theorem refun0

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

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
2		eqid 2738	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
3	1, 2	refbas 22661	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
4	3	adantr 481	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∪ 𝐵 = ∪ 𝐴)
5		elun 4083	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ {∅}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {∅}))
6		refssex 22662	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
7	6	adantlr 712	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8		0ss 4330	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑦
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝐴Ref𝐵 ∧ 𝑦 ∈ 𝐵) → ∅ ⊆ 𝑦)
10	9	reximdva0 4285	. . . . . . 7 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦)
11	10	adantr 481	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦)
12		elsni 4578	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
13		sseq1 3946	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
14	13	rexbidv 3226	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦))
15	12, 14	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦))
16	15	adantl 482	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦))
17	11, 16	mpbird 256	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
18	7, 17	jaodan 955	. . . 4 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {∅})) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
19	5, 18	sylan2b 594	. . 3 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ (𝐴 ∪ {∅})) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
20	19	ralrimiva 3103	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
21		refrel 22659	. . . . . 6 ⊢ Rel Ref
22	21	brrelex1i 5643	. . . . 5 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
23		p0ex 5307	. . . . 5 ⊢ {∅} ∈ V
24		unexg 7599	. . . . 5 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ∪ {∅}) ∈ V)
25	22, 23, 24	sylancl 586	. . . 4 ⊢ (𝐴Ref𝐵 → (𝐴 ∪ {∅}) ∈ V)
26		uniun 4864	. . . . . 6 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
27		0ex 5231	. . . . . . . 8 ⊢ ∅ ∈ V
28	27	unisn 4861	. . . . . . 7 ⊢ ∪ {∅} = ∅
29	28	uneq2i 4094	. . . . . 6 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
30		un0 4324	. . . . . 6 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
31	26, 29, 30	3eqtrri 2771	. . . . 5 ⊢ ∪ 𝐴 = ∪ (𝐴 ∪ {∅})
32	31, 2	isref 22660	. . . 4 ⊢ ((𝐴 ∪ {∅}) ∈ V → ((𝐴 ∪ {∅})Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
33	25, 32	syl 17	. . 3 ⊢ (𝐴Ref𝐵 → ((𝐴 ∪ {∅})Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
34	33	adantr 481	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ((𝐴 ∪ {∅})Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
35	4, 20, 34	mpbir2and 710	1 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561  ∪ 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-ne 2944  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