Theorem undom 5497

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		unen 5493	. . . . . . . . . . . . . . 15 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/))) -> (A u. (C \ A)) ~~ (x u. y))
2		undif2 2950	. . . . . . . . . . . . . . 15 |- (A u. (C \ A)) = (A u. C)
3	1, 2	syl5eqbrr 3371	. . . . . . . . . . . . . 14 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/))) -> (A u. C) ~~ (x u. y))
4		sseq2 2639	. . . . . . . . . . . . . . . . . 18 |- ((B i^i D) = (/) -> ((x i^i y) C_ (B i^i D) <-> (x i^i y) C_ (/)))
5		ss0b 2901	. . . . . . . . . . . . . . . . . 18 |- ((x i^i y) C_ (/) <-> (x i^i y) = (/))
6	4, 5	syl6bb 595	. . . . . . . . . . . . . . . . 17 |- ((B i^i D) = (/) -> ((x i^i y) C_ (B i^i D) <-> (x i^i y) = (/)))
7		ss2in 2820	. . . . . . . . . . . . . . . . 17 |- ((x C_ B /\ y C_ D) -> (x i^i y) C_ (B i^i D))
8	6, 7	syl5bi 225	. . . . . . . . . . . . . . . 16 |- ((B i^i D) = (/) -> ((x C_ B /\ y C_ D) -> (x i^i y) = (/)))
9	8	imp 377	. . . . . . . . . . . . . . 15 |- (((B i^i D) = (/) /\ (x C_ B /\ y C_ D)) -> (x i^i y) = (/))
10		difdisj 2945	. . . . . . . . . . . . . . 15 |- (A i^i (C \ A)) = (/)
11	9, 10	jctil 316	. . . . . . . . . . . . . 14 |- (((B i^i D) = (/) /\ (x C_ B /\ y C_ D)) -> ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/)))
12	3, 11	sylan2 500	. . . . . . . . . . . . 13 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((B i^i D) = (/) /\ (x C_ B /\ y C_ D))) -> (A u. C) ~~ (x u. y))
13	12	anassrs 489	. . . . . . . . . . . 12 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (B i^i D) = (/)) /\ (x C_ B /\ y C_ D)) -> (A u. C) ~~ (x u. y))
14	13	an1rs 547	. . . . . . . . . . 11 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x C_ B /\ y C_ D)) /\ (B i^i D) = (/)) -> (A u. C) ~~ (x u. y))
15		unss12 2778	. . . . . . . . . . . . 13 |- ((x C_ B /\ y C_ D) -> (x u. y) C_ (B u. D))
16		undom.1	. . . . . . . . . . . . . . 15 |- B e. _V
17		undom.3	. . . . . . . . . . . . . . 15 |- D e. _V
18	16, 17	unex 3796	. . . . . . . . . . . . . 14 |- (B u. D) e. _V
19		ssdom2g 5468	. . . . . . . . . . . . . 14 |- ((B u. D) e. _V -> ((x u. y) C_ (B u. D) -> (x u. y) ~<_ (B u. D)))
20	18, 19	ax-mp 7	. . . . . . . . . . . . 13 |- ((x u. y) C_ (B u. D) -> (x u. y) ~<_ (B u. D))
21	15, 20	syl 12	. . . . . . . . . . . 12 |- ((x C_ B /\ y C_ D) -> (x u. y) ~<_ (B u. D))
22	21	ad2antlr 441	. . . . . . . . . . 11 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x C_ B /\ y C_ D)) /\ (B i^i D) = (/)) -> (x u. y) ~<_ (B u. D))
23		endomtr 5479	. . . . . . . . . . 11 |- (((A u. C) ~~ (x u. y) /\ (x u. y) ~<_ (B u. D)) -> (A u. C) ~<_ (B u. D))
24	14, 22, 23	syl11anc 524	. . . . . . . . . 10 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x C_ B /\ y C_ D)) /\ (B i^i D) = (/)) -> (A u. C) ~<_ (B u. D))
25	24	ex 402	. . . . . . . . 9 |- (((A ~~ x /\ (C \ A) ~~ y) /\ (x C_ B /\ y C_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
26	25	an4s 566	. . . . . . . 8 |- (((A ~~ x /\ x C_ B) /\ ((C \ A) ~~ y /\ y C_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
27	26	ex 402	. . . . . . 7 |- ((A ~~ x /\ x C_ B) -> (((C \ A) ~~ y /\ y C_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
28	27	19.23aiv 1674	. . . . . 6 |- (E.x(A ~~ x /\ x C_ B) -> (((C \ A) ~~ y /\ y C_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
29	28	19.23adv 1584	. . . . 5 |- (E.x(A ~~ x /\ x C_ B) -> (E.y((C \ A) ~~ y /\ y C_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
30	29	imp 377	. . . 4 |- ((E.x(A ~~ x /\ x C_ B) /\ E.y((C \ A) ~~ y /\ y C_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
31	16	domen 5438	. . . 4 |- (A ~<_ B <-> E.x(A ~~ x /\ x C_ B))
32	17	domen 5438	. . . 4 |- ((C \ A) ~<_ D <-> E.y((C \ A) ~~ y /\ y C_ D))
33	30, 31, 32	syl2anb 504	. . 3 |- ((A ~<_ B /\ (C \ A) ~<_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
34		undom.2	. . . . 5 |- C e. _V
35		difss 2735	. . . . 5 |- (C \ A) C_ C
36		ssdom2g 5468	. . . . 5 |- (C e. _V -> ((C \ A) C_ C -> (C \ A) ~<_ C))
37	34, 35, 36	mp2 54	. . . 4 |- (C \ A) ~<_ C
38		domtr 5474	. . . 4 |- (((C \ A) ~<_ C /\ C ~<_ D) -> (C \ A) ~<_ D)
39	37, 38	mpan 759	. . 3 |- (C ~<_ D -> (C \ A) ~<_ D)
40	33, 39	sylan2 500	. 2 |- ((A ~<_ B /\ C ~<_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
41	40	imp 377	1 |- (((A ~<_ B /\ C ~<_ D) /\ (B i^i D) = (/)) -> (A u. C) ~<_ (B u. D))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875   class class class wbr 3338   ~~ cen 5423   ~<_ cdom 5424

This theorem is referenced by:  fodomfi 5656  unxpdom2 5997  sucxpdom 5998  uncdadom 6069  cdadom1 6083

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-en 5427  df-dom 5428

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