Theorem refun0 23653

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 2763	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
2		eqid 2763	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
3	1, 2	refbas 23648	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
4	3	adantr 485	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∪ 𝐵 = ∪ 𝐴)
5		elun 4108	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ {∅}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {∅}))
6		refssex 23649	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
7	6	adantlr 727	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8		0ss 4358	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑦
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝐴Ref𝐵 ∧ 𝑦 ∈ 𝐵) → ∅ ⊆ 𝑦)
10	9	reximdva0 4311	. . . . . . 7 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦)
11	10	adantr 485	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦)
12		elsni 4607	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
13		sseq1 3963	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
14	13	rexbidv 3189	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦))
15	12, 14	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦))
16	15	adantl 486	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦))
17	11, 16	mpbird 260	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
18	7, 17	jaodan 972	. . . 4 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {∅})) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
19	5, 18	sylan2b 605	. . 3 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ (𝐴 ∪ {∅})) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
20	19	ralrimiva 3157	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
21		refrel 23646	. . . . . 6 ⊢ Rel Ref
22	21	brrelex1i 5719	. . . . 5 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
23		p0ex 5357	. . . . 5 ⊢ {∅} ∈ V
24		unexg 7743	. . . . 5 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ∪ {∅}) ∈ V)
25	22, 23, 24	sylancl 597	. . . 4 ⊢ (𝐴Ref𝐵 → (𝐴 ∪ {∅}) ∈ V)
26		uniun 4896	. . . . . 6 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
27		0ex 5271	. . . . . . . 8 ⊢ ∅ ∈ V
28	27	unisn 4892	. . . . . . 7 ⊢ ∪ {∅} = ∅
29	28	uneq2i 4120	. . . . . 6 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
30		un0 4352	. . . . . 6 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
31	26, 29, 30	3eqtrri 2791	. . . . 5 ⊢ ∪ 𝐴 = ∪ (𝐴 ∪ {∅})
32	31, 2	isref 23647	. . . 4 ⊢ ((𝐴 ∪ {∅}) ∈ V → ((𝐴 ∪ {∅})Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
33	25, 32	syl 18	. . 3 ⊢ (𝐴Ref𝐵 → ((𝐴 ∪ {∅})Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
34	33	adantr 485	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ((𝐴 ∪ {∅})Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
35	4, 20, 34	mpbir2and 725	1 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  {csn 4590  ∪ cuni 4873   class class class wbr 5110  Refcref 23640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-ref 23643

This theorem is referenced by:  locfinref  34209