Theorem refun0 23490

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 2737	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
2		eqid 2737	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝐵
3	1, 2	refbas 23485	. . 3 ⊢ (𝐴Ref𝐵 → ∪ 𝐵 = ∪ 𝐴)
4	3	adantr 480	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∪ 𝐵 = ∪ 𝐴)
5		elun 4094	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ {∅}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {∅}))
6		refssex 23486	. . . . . 6 ⊢ ((𝐴Ref𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
7	6	adantlr 716	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
8		0ss 4341	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑦
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝐴Ref𝐵 ∧ 𝑦 ∈ 𝐵) → ∅ ⊆ 𝑦)
10	9	reximdva0 4296	. . . . . . 7 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦)
11	10	adantr 480	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦)
12		elsni 4585	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
13		sseq1 3948	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
14	13	rexbidv 3162	. . . . . . . 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 481	. . . . . 6 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 ∅ ⊆ 𝑦))
17	11, 16	mpbird 257	. . . . 5 ⊢ (((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ {∅}) → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
18	7, 17	jaodan 960	. . . 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 3130	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
21		refrel 23483	. . . . . 6 ⊢ Rel Ref
22	21	brrelex1i 5680	. . . . 5 ⊢ (𝐴Ref𝐵 → 𝐴 ∈ V)
23		p0ex 5321	. . . . 5 ⊢ {∅} ∈ V
24		unexg 7690	. . . . 5 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ∪ {∅}) ∈ V)
25	22, 23, 24	sylancl 587	. . . 4 ⊢ (𝐴Ref𝐵 → (𝐴 ∪ {∅}) ∈ V)
26		uniun 4874	. . . . . 6 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
27		0ex 5242	. . . . . . . 8 ⊢ ∅ ∈ V
28	27	unisn 4870	. . . . . . 7 ⊢ ∪ {∅} = ∅
29	28	uneq2i 4106	. . . . . 6 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
30		un0 4335	. . . . . 6 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
31	26, 29, 30	3eqtrri 2765	. . . . 5 ⊢ ∪ 𝐴 = ∪ (𝐴 ∪ {∅})
32	31, 2	isref 23484	. . . 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 480	. 2 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → ((𝐴 ∪ {∅})Ref𝐵 ↔ (∪ 𝐵 = ∪ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∪ {∅})∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
35	4, 20, 34	mpbir2and 714	1 ⊢ ((𝐴Ref𝐵 ∧ 𝐵 ≠ ∅) → (𝐴 ∪ {∅})Ref𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851   class class class wbr 5086  Refcref 23477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-ref 23480

This theorem is referenced by:  locfinref  34001