Theorem undom 3342

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

Hypotheses

Ref	Expression
undom.1	|- B e. V
undom.2	|- C e. V
undom.3	|- D e. V

Assertion

Ref	Expression
undom	|- (((A ~<_ B /\ C ~<_ D) /\ (B i^i D) = (/)) -> (A u. C) ~<_ (B u. D))

Proof of Theorem undom

Step	Hyp	Ref	Expression
1		endomtr 3325	. . . . . . . . . . 11 |- (((A u. C) ~~ (x u. y) /\ (x u. y) ~<_ (B u. D)) -> (A u. C) ~<_ (B u. D))
2		unen 3338	. . . . . . . . . . . . . . 15 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/))) -> (A u. (C \ A)) ~~ (x u. y))
3		undif2 1762	. . . . . . . . . . . . . . . 16 |- (A u. (C \ A)) = (A u. C)
4	3	cleqcomi 1105	. . . . . . . . . . . . . . 15 |- (A u. C) = (A u. (C \ A))
5	2, 4	syl5eqbr 2089	. . . . . . . . . . . . . 14 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/))) -> (A u. C) ~~ (x u. y))
6		sseq2 1522	. . . . . . . . . . . . . . . . . 18 |- ((B i^i D) = (/) -> ((x i^i y) (_ (B i^i D) <-> (x i^i y) (_ (/)))
7		ss0b 1726	. . . . . . . . . . . . . . . . . 18 |- ((x i^i y) (_ (/) <-> (x i^i y) = (/))
8	6, 7	syl6bb 414	. . . . . . . . . . . . . . . . 17 |- ((B i^i D) = (/) -> ((x i^i y) (_ (B i^i D) <-> (x i^i y) = (/)))
9		ss2in 1663	. . . . . . . . . . . . . . . . 17 |- ((x (_ B /\ y (_ D) -> (x i^i y) (_ (B i^i D))
10	8, 9	syl5bi 183	. . . . . . . . . . . . . . . 16 |- ((B i^i D) = (/) -> ((x (_ B /\ y (_ D) -> (x i^i y) = (/)))
11	10	imp 277	. . . . . . . . . . . . . . 15 |- (((B i^i D) = (/) /\ (x (_ B /\ y (_ D)) -> (x i^i y) = (/))
12		difdisj 1758	. . . . . . . . . . . . . . 15 |- (A i^i (C \ A)) = (/)
13	11, 12	jctil 240	. . . . . . . . . . . . . 14 |- (((B i^i D) = (/) /\ (x (_ B /\ y (_ D)) -> ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/)))
14	5, 13	sylan2 346	. . . . . . . . . . . . 13 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((B i^i D) = (/) /\ (x (_ B /\ y (_ D))) -> (A u. C) ~~ (x u. y))
15	14	anassrs 338	. . . . . . . . . . . 12 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (B i^i D) = (/)) /\ (x (_ B /\ y (_ D)) -> (A u. C) ~~ (x u. y))
16	15	an1rs 373	. . . . . . . . . . 11 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) /\ (B i^i D) = (/)) -> (A u. C) ~~ (x u. y))
17		unss12 1630	. . . . . . . . . . . . 13 |- ((x (_ B /\ y (_ D) -> (x u. y) (_ (B u. D))
18		undom.1	. . . . . . . . . . . . . . 15 |- B e. V
19		undom.3	. . . . . . . . . . . . . . 15 |- D e. V
20	18, 19	unex 1949	. . . . . . . . . . . . . 14 |- (B u. D) e. V
21		ssdom2g 3312	. . . . . . . . . . . . . 14 |- ((B u. D) e. V -> ((x u. y) (_ (B u. D) -> (x u. y) ~<_ (B u. D)))
22	20, 21	ax-mp 6	. . . . . . . . . . . . 13 |- ((x u. y) (_ (B u. D) -> (x u. y) ~<_ (B u. D))
23	17, 22	syl 12	. . . . . . . . . . . 12 |- ((x (_ B /\ y (_ D) -> (x u. y) ~<_ (B u. D))
24	23	ad2antlr 321	. . . . . . . . . . 11 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) /\ (B i^i D) = (/)) -> (x u. y) ~<_ (B u. D))
25	1, 16, 24	sylanc 361	. . . . . . . . . 10 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) /\ (B i^i D) = (/)) -> (A u. C) ~<_ (B u. D))
26	25	exp 291	. . . . . . . . 9 |- (((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
27	26	an4s 390	. . . . . . . 8 |- (((A ~~ x /\ x (_ B) /\ ((C \ A) ~~ y /\ y (_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
28	27	exp 291	. . . . . . 7 |- ((A ~~ x /\ x (_ B) -> (((C \ A) ~~ y /\ y (_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
29	28	19.23aiv 952	. . . . . 6 |- (E.x(A ~~ x /\ x (_ B) -> (((C \ A) ~~ y /\ y (_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
30	29	19.23adv 954	. . . . 5 |- (E.x(A ~~ x /\ x (_ B) -> (E.y((C \ A) ~~ y /\ y (_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
31	30	imp 277	. . . 4 |- ((E.x(A ~~ x /\ x (_ B) /\ E.y((C \ A) ~~ y /\ y (_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
32	18	domen 3284	. . . 4 |- (A ~<_ B <-> E.x(A ~~ x /\ x (_ B))
33	19	domen 3284	. . . 4 |- ((C \ A) ~<_ D <-> E.y((C \ A) ~~ y /\ y (_ D))
34	31, 32, 33	syl2anb 350	. . 3 |- ((A ~<_ B /\ (C \ A) ~<_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
35		undom.2	. . . . 5 |- C e. V
36		difss 1596	. . . . 5 |- (C \ A) (_ C
37		ssdom2g 3312	. . . . 5 |- (C e. V -> ((C \ A) (_ C -> (C \ A) ~<_ C))
38	35, 36, 37	mp2 43	. . . 4 |- (C \ A) ~<_ C
39		domtr 3320	. . . 4 |- (((C \ A) ~<_ C /\ C ~<_ D) -> (C \ A) ~<_ D)
40	38, 39	mpan 518	. . 3 |- (C ~<_ D -> (C \ A) ~<_ D)
41	34, 40	sylan2 346	. 2 |- ((A ~<_ B /\ C ~<_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
42	41	imp 277	1 |- (((A ~<_ B /\ C ~<_ D) /\ (B i^i D) = (/)) -> (A u. C) ~<_ (B u. D))

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348   \ cdif 1484   u. cun 1485   i^i cin 1486   (_ wss 1487  (/)c0 1707   class class class wbr 2054   ~~ cen 3271   ~<_ cdom 3272

This theorem is referenced by:  unxpdom2 3651  sucxpdom 3652  uncdadom 3718  cdadom1 3727

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274  df-dom 3275

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