Theorem refun0 22125

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 2823	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
2		eqid 2823	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
3	1, 2	refbas 22120	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
4	3	adantr 483	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∪ 𝐵 = ∪ 𝐴)
5		elun 4127	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ {∅}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {∅}))
6		refssex 22121	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
7	6	adantlr 713	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8		0ss 4352	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑦
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝐴Ref𝐵 ∧ 𝑦 ∈ 𝐵) → ∅ ⊆ 𝑦)
10	9	reximdva0 4314	. . . . . . 7 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦)
11	10	adantr 483	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦)
12		elsni 4586	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
13		sseq1 3994	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
14	13	rexbidv 3299	. . . . . . . 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 484	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦))
17	11, 16	mpbird 259	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
18	7, 17	jaodan 954	. . . 4 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {∅})) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
19	5, 18	sylan2b 595	. . 3 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ (𝐴 ∪ {∅})) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
20	19	ralrimiva 3184	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
21		refrel 22118	. . . . . 6 ⊢ Rel Ref
22	21	brrelex1i 5610	. . . . 5 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
23		p0ex 5287	. . . . 5 ⊢ {∅} ∈ V
24		unexg 7474	. . . . 5 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ∪ {∅}) ∈ V)
25	22, 23, 24	sylancl 588	. . . 4 ⊢ (𝐴Ref𝐵 → (𝐴 ∪ {∅}) ∈ V)
26		uniun 4863	. . . . . 6 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
27		0ex 5213	. . . . . . . 8 ⊢ ∅ ∈ V
28	27	unisn 4860	. . . . . . 7 ⊢ ∪ {∅} = ∅
29	28	uneq2i 4138	. . . . . 6 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
30		un0 4346	. . . . . 6 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
31	26, 29, 30	3eqtrri 2851	. . . . 5 ⊢ ∪ 𝐴 = ∪ (𝐴 ∪ {∅})
32	31, 2	isref 22119	. . . 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 483	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ((𝐴 ∪ {∅})Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
35	4, 20, 34	mpbir2and 711	1 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∪ cun 3936   ⊆ wss 3938  ∅c0 4293  {csn 4569  ∪ cuni 4840   class class class wbr 5068  Refcref 22112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-ref 22115

This theorem is referenced by:  locfinref  31107