Theorem undom 4444

Description: Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257.

Hypotheses

Ref	Expression
undom.1	⊢ B ∈ V
undom.2	⊢ C ∈ V
undom.3	⊢ D ∈ V

Assertion

Ref	Expression
undom	⊢ (((A ≼ B ⋀ C ≼ D) ⋀ (B ∩ D) = ∅) → (A ∪ C) ≼ (B ∪ D))

Proof of Theorem undom

Step	Hyp	Ref	Expression
1		endomtr 4426	. . . . . . . . . . 11 ⊢ (((A ∪ C) ≈ (x ∪ y) ⋀ (x ∪ y) ≼ (B ∪ D)) → (A ∪ C) ≼ (B ∪ D))
2		unen 4440	. . . . . . . . . . . . . . 15 ⊢ (((A ≈ x ⋀ (C ∖ A) ≈ y) ⋀ ((A ∩ (C ∖ A)) = ∅ ⋀ (x ∩ y) = ∅)) → (A ∪ (C ∖ A)) ≈ (x ∪ y))
3		undif2 2345	. . . . . . . . . . . . . . 15 ⊢ (A ∪ (C ∖ A)) = (A ∪ C)
4	2, 3	syl5eqbrr 2654	. . . . . . . . . . . . . 14 ⊢ (((A ≈ x ⋀ (C ∖ A) ≈ y) ⋀ ((A ∩ (C ∖ A)) = ∅ ⋀ (x ∩ y) = ∅)) → (A ∪ C) ≈ (x ∪ y))
5		sseq2 2086	. . . . . . . . . . . . . . . . . 18 ⊢ ((B ∩ D) = ∅ → ((x ∩ y) ⊆ (B ∩ D) ↔ (x ∩ y) ⊆ ∅))
6		ss0b 2306	. . . . . . . . . . . . . . . . . 18 ⊢ ((x ∩ y) ⊆ ∅ ↔ (x ∩ y) = ∅)
7	5, 6	syl6bb 538	. . . . . . . . . . . . . . . . 17 ⊢ ((B ∩ D) = ∅ → ((x ∩ y) ⊆ (B ∩ D) ↔ (x ∩ y) = ∅))
8		ss2in 2239	. . . . . . . . . . . . . . . . 17 ⊢ ((x ⊆ B ⋀ y ⊆ D) → (x ∩ y) ⊆ (B ∩ D))
9	7, 8	syl5bi 208	. . . . . . . . . . . . . . . 16 ⊢ ((B ∩ D) = ∅ → ((x ⊆ B ⋀ y ⊆ D) → (x ∩ y) = ∅))
10	9	imp 350	. . . . . . . . . . . . . . 15 ⊢ (((B ∩ D) = ∅ ⋀ (x ⊆ B ⋀ y ⊆ D)) → (x ∩ y) = ∅)
11		difdisj 2341	. . . . . . . . . . . . . . 15 ⊢ (A ∩ (C ∖ A)) = ∅
12	10, 11	jctil 292	. . . . . . . . . . . . . 14 ⊢ (((B ∩ D) = ∅ ⋀ (x ⊆ B ⋀ y ⊆ D)) → ((A ∩ (C ∖ A)) = ∅ ⋀ (x ∩ y) = ∅))
13	4, 12	sylan2 453	. . . . . . . . . . . . 13 ⊢ (((A ≈ x ⋀ (C ∖ A) ≈ y) ⋀ ((B ∩ D) = ∅ ⋀ (x ⊆ B ⋀ y ⊆ D))) → (A ∪ C) ≈ (x ∪ y))
14	13	anassrs 443	. . . . . . . . . . . 12 ⊢ ((((A ≈ x ⋀ (C ∖ A) ≈ y) ⋀ (B ∩ D) = ∅) ⋀ (x ⊆ B ⋀ y ⊆ D)) → (A ∪ C) ≈ (x ∪ y))
15	14	an1rs 491	. . . . . . . . . . 11 ⊢ ((((A ≈ x ⋀ (C ∖ A) ≈ y) ⋀ (x ⊆ B ⋀ y ⊆ D)) ⋀ (B ∩ D) = ∅) → (A ∪ C) ≈ (x ∪ y))
16		unss12 2205	. . . . . . . . . . . . 13 ⊢ ((x ⊆ B ⋀ y ⊆ D) → (x ∪ y) ⊆ (B ∪ D))
17		undom.1	. . . . . . . . . . . . . . 15 ⊢ B ∈ V
18		undom.3	. . . . . . . . . . . . . . 15 ⊢ D ∈ V
19	17, 18	unex 2878	. . . . . . . . . . . . . 14 ⊢ (B ∪ D) ∈ V
20		ssdom2g 4415	. . . . . . . . . . . . . 14 ⊢ ((B ∪ D) ∈ V → ((x ∪ y) ⊆ (B ∪ D) → (x ∪ y) ≼ (B ∪ D)))
21	19, 20	ax-mp 7	. . . . . . . . . . . . 13 ⊢ ((x ∪ y) ⊆ (B ∪ D) → (x ∪ y) ≼ (B ∪ D))
22	16, 21	syl 10	. . . . . . . . . . . 12 ⊢ ((x ⊆ B ⋀ y ⊆ D) → (x ∪ y) ≼ (B ∪ D))
23	22	ad2antlr 407	. . . . . . . . . . 11 ⊢ ((((A ≈ x ⋀ (C ∖ A) ≈ y) ⋀ (x ⊆ B ⋀ y ⊆ D)) ⋀ (B ∩ D) = ∅) → (x ∪ y) ≼ (B ∪ D))
24	1, 15, 23	sylanc 473	. . . . . . . . . 10 ⊢ ((((A ≈ x ⋀ (C ∖ A) ≈ y) ⋀ (x ⊆ B ⋀ y ⊆ D)) ⋀ (B ∩ D) = ∅) → (A ∪ C) ≼ (B ∪ D))
25	24	ex 373	. . . . . . . . 9 ⊢ (((A ≈ x ⋀ (C ∖ A) ≈ y) ⋀ (x ⊆ B ⋀ y ⊆ D)) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
26	25	an4s 510	. . . . . . . 8 ⊢ (((A ≈ x ⋀ x ⊆ B) ⋀ ((C ∖ A) ≈ y ⋀ y ⊆ D)) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
27	26	ex 373	. . . . . . 7 ⊢ ((A ≈ x ⋀ x ⊆ B) → (((C ∖ A) ≈ y ⋀ y ⊆ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D))))
28	27	19.23aiv 1297	. . . . . 6 ⊢ (∃x(A ≈ x ⋀ x ⊆ B) → (((C ∖ A) ≈ y ⋀ y ⊆ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D))))
29	28	19.23adv 1216	. . . . 5 ⊢ (∃x(A ≈ x ⋀ x ⊆ B) → (∃y((C ∖ A) ≈ y ⋀ y ⊆ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D))))
30	29	imp 350	. . . 4 ⊢ ((∃x(A ≈ x ⋀ x ⊆ B) ⋀ ∃y((C ∖ A) ≈ y ⋀ y ⊆ D)) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
31	17	domen 4385	. . . 4 ⊢ (A ≼ B ↔ ∃x(A ≈ x ⋀ x ⊆ B))
32	18	domen 4385	. . . 4 ⊢ ((C ∖ A) ≼ D ↔ ∃y((C ∖ A) ≈ y ⋀ y ⊆ D))
33	30, 31, 32	syl2anb 457	. . 3 ⊢ ((A ≼ B ⋀ (C ∖ A) ≼ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
34		undom.2	. . . . 5 ⊢ C ∈ V
35		difss 2170	. . . . 5 ⊢ (C ∖ A) ⊆ C
36		ssdom2g 4415	. . . . 5 ⊢ (C ∈ V → ((C ∖ A) ⊆ C → (C ∖ A) ≼ C))
37	34, 35, 36	mp2 43	. . . 4 ⊢ (C ∖ A) ≼ C
38		domtr 4421	. . . 4 ⊢ (((C ∖ A) ≼ C ⋀ C ≼ D) → (C ∖ A) ≼ D)
39	37, 38	mpan 697	. . 3 ⊢ (C ≼ D → (C ∖ A) ≼ D)
40	33, 39	sylan2 453	. 2 ⊢ ((A ≼ B ⋀ C ≼ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
41	40	imp 350	1 ⊢ (((A ≼ B ⋀ C ≼ D) ⋀ (B ∩ D) = ∅) → (A ∪ C) ≼ (B ∪ D))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃wex 982  Vcvv 1814   ∖ cdif 2047   ∪ cun 2048   ∩ cin 2049   ⊆ wss 2050  ∅c0 2283   class class class wbr 2624   ≈ cen 4370   ≼ cdom 4371

This theorem is referenced by:  fodomfi 4575  fodomfiOLD 4576  unxpdom2 4856  sucxpdom 4857  uncdadom 4933  cdadom1 4945

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-en 4374  df-dom 4375

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