Theorem undom 4418

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 4401	. . . . . . . . . . 11 |- (((A u. C) ~~ (x u. y) /\ (x u. y) ~<_ (B u. D)) -> (A u. C) ~<_ (B u. D))
2		unen 4414	. . . . . . . . . . . . . . 15 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/))) -> (A u. (C \ A)) ~~ (x u. y))
3		undif2 2331	. . . . . . . . . . . . . . 15 |- (A u. (C \ A)) = (A u. C)
4	2, 3	syl5eqbrr 2639	. . . . . . . . . . . . . 14 |- (((A ~~ x /\ (C \ A) ~~ y) /\ ((A i^i (C \ A)) = (/) /\ (x i^i y) = (/))) -> (A u. C) ~~ (x u. y))
5		sseq2 2073	. . . . . . . . . . . . . . . . . 18 |- ((B i^i D) = (/) -> ((x i^i y) (_ (B i^i D) <-> (x i^i y) (_ (/)))
6		ss0b 2292	. . . . . . . . . . . . . . . . . 18 |- ((x i^i y) (_ (/) <-> (x i^i y) = (/))
7	5, 6	syl6bb 534	. . . . . . . . . . . . . . . . 17 |- ((B i^i D) = (/) -> ((x i^i y) (_ (B i^i D) <-> (x i^i y) = (/)))
8		ss2in 2226	. . . . . . . . . . . . . . . . 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 2327	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . 11 |- ((((A ~~ x /\ (C \ A) ~~ y) /\ (x (_ B /\ y (_ D)) /\ (B i^i D) = (/)) -> (A u. C) ~~ (x u. y))
16		unss12 2192	. . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . 14 |- (B u. D) e. V
20		ssdom2g 4390	. . . . . . . . . . . . . 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 507	. . . . . . . 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 1290	. . . . . 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 1209	. . . . 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 4361	. . . 4 |- (A ~<_ B <-> E.x(A ~~ x /\ x (_ B))
32	18	domen 4361	. . . 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 2157	. . . . 5 |- (C \ A) (_ C
36		ssdom2g 4390	. . . . 5 |- (C e. V -> ((C \ A) (_ C -> (C \ A) ~<_ C))
37	34, 35, 36	mp2 43	. . . 4 |- (C \ A) ~<_ C
38		domtr 4396	. . . 4 |- (((C \ A) ~<_ C /\ C ~<_ D) -> (C \ A) ~<_ D)
39	37, 38	mpan 693	. . 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 953   e. wcel 955  E.wex 977  Vcvv 1802   \ cdif 2034   u. cun 2035   i^i cin 2036   (_ wss 2037  (/)c0 2270   class class class wbr 2609   ~~ cen 4348   ~<_ cdom 4349

This theorem is referenced by:  fodomfi 4540  unxpdom2 4817  sucxpdom 4818  uncdadom 4893  cdadom1 4905

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-en 4351  df-dom 4352

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