Theorem undom 4438

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 4420	. . . . . . . . . . 11 |- (((A u. C) ~~ (x u. y) /\ (x u. y) ~<_ (B u. D)) -> (A u. C) ~<_ (B u. D))
2		unen 4434	. . . . . . . . . . . . . . 15 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/))) -> (A u. (C \ A)) ~~ (x u. y))
3		undif2 2341	. . . . . . . . . . . . . . 15 |- (A u. (C \ A)) = (A u. C)
4	2, 3	syl5eqbrr 2649	. . . . . . . . . . . . . 14 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/))) -> (A u. C) ~~ (x u. y))
5		sseq2 2083	. . . . . . . . . . . . . . . . . 18 |- ((B i^i D) = (/) -> ((x i^i y) (_ (B i^i D) <-> (x i^i y) (_ (/)))
6		ss0b 2302	. . . . . . . . . . . . . . . . . 18 |- ((x i^i y) (_ (/) <-> (x i^i y) = (/))
7	5, 6	syl6bb 536	. . . . . . . . . . . . . . . . 17 |- ((B i^i D) = (/) -> ((x i^i y) (_ (B i^i D) <-> (x i^i y) = (/)))
8		ss2in 2236	. . . . . . . . . . . . . . . . 17 |- ((x (_ B /\ y (_ D) -> (x i^i y) (_ (B i^i D))
9	7, 8	syl5bi 208	. . . . . . . . . . . . . . . 16 |- ((B i^i D) = (/) -> ((x (_ B /\ y (_ D) -> (x i^i y) = (/)))
10	9	imp 350	. . . . . . . . . . . . . . 15 |- (((B i^i D) = (/) /\ (x (_ B /\ y (_ D)) -> (x i^i y) = (/))
11		difdisj 2337	. . . . . . . . . . . . . . 15 |- (A i^i (C \ A)) = (/)
12	10, 11	jctil 292	. . . . . . . . . . . . . 14 |- (((B i^i D) = (/) /\ (x (_ B /\ y (_ D)) -> ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/)))
13	4, 12	sylan2 451	. . . . . . . . . . . . 13 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((B i^i D) = (/) /\ (x (_ B /\ y (_ D))) -> (A u. C) ~~ (x u. y))
14	13	anassrs 441	. . . . . . . . . . . 12 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (B i^i D) = (/)) /\ (x (_ B /\ y (_ D)) -> (A u. C) ~~ (x u. y))
15	14	an1rs 489	. . . . . . . . . . 11 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) /\ (B i^i D) = (/)) -> (A u. C) ~~ (x u. y))
16		unss12 2202	. . . . . . . . . . . . 13 |- ((x (_ B /\ y (_ D) -> (x u. y) (_ (B u. D))
17		undom.1	. . . . . . . . . . . . . . 15 |- B e. V
18		undom.3	. . . . . . . . . . . . . . 15 |- D e. V
19	17, 18	unex 2872	. . . . . . . . . . . . . 14 |- (B u. D) e. V
20		ssdom2g 4409	. . . . . . . . . . . . . 14 |- ((B u. D) e. V -> ((x u. y) (_ (B u. D) -> (x u. y) ~<_ (B u. D)))
21	19, 20	ax-mp 7	. . . . . . . . . . . . 13 |- ((x u. y) (_ (B u. D) -> (x u. y) ~<_ (B u. D))
22	16, 21	syl 10	. . . . . . . . . . . 12 |- ((x (_ B /\ y (_ D) -> (x u. y) ~<_ (B u. D))
23	22	ad2antlr 405	. . . . . . . . . . 11 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) /\ (B i^i D) = (/)) -> (x u. y) ~<_ (B u. D))
24	1, 15, 23	sylanc 471	. . . . . . . . . 10 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) /\ (B i^i D) = (/)) -> (A u. C) ~<_ (B u. D))
25	24	ex 373	. . . . . . . . 9 |- (((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
26	25	an4s 508	. . . . . . . 8 |- (((A ~~ x /\ x (_ B) /\ ((C \ A) ~~ y /\ y (_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
27	26	ex 373	. . . . . . 7 |- ((A ~~ x /\ x (_ B) -> (((C \ A) ~~ y /\ y (_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
28	27	19.23aiv 1295	. . . . . 6 |- (E.x(A ~~ x /\ x (_ B) -> (((C \ A) ~~ y /\ y (_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
29	28	19.23adv 1214	. . . . 5 |- (E.x(A ~~ x /\ x (_ B) -> (E.y((C \ A) ~~ y /\ y (_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D))))
30	29	imp 350	. . . 4 |- ((E.x(A ~~ x /\ x (_ B) /\ E.y((C \ A) ~~ y /\ y (_ D)) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
31	17	domen 4379	. . . 4 |- (A ~<_ B <-> E.x(A ~~ x /\ x (_ B))
32	18	domen 4379	. . . 4 |- ((C \ A) ~<_ D <-> E.y((C \ A) ~~ y /\ y (_ D))
33	30, 31, 32	syl2anb 455	. . 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 2167	. . . . 5 |- (C \ A) (_ C
36		ssdom2g 4409	. . . . 5 |- (C e. V -> ((C \ A) (_ C -> (C \ A) ~<_ C))
37	34, 35, 36	mp2 43	. . . 4 |- (C \ A) ~<_ C
38		domtr 4415	. . . 4 |- (((C \ A) ~<_ C /\ C ~<_ D) -> (C \ A) ~<_ D)
39	37, 38	mpan 695	. . 3 |- (C ~<_ D -> (C \ A) ~<_ D)
40	33, 39	sylan2 451	. 2 |- ((A ~<_ B /\ C ~<_ D) -> ((B i^i D) = (/) -> (A u. C) ~<_ (B u. D)))
41	40	imp 350	1 |- (((A ~<_ B /\ C ~<_ D) /\ (B i^i D) = (/)) -> (A u. C) ~<_ (B u. D))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811   \ cdif 2044   u. cun 2045   i^i cin 2046   (_ wss 2047  (/)c0 2280   class class class wbr 2619   ~~ cen 4364   ~<_ cdom 4365

This theorem is referenced by:  fodomfiOLD 4566  unxpdom2 4845  sucxpdom 4846  uncdadom 4921  cdadom1 4933

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368  df-dom 4369

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