Theorem unxpdom 4816

Description: Cross product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93.

Assertion

Ref	Expression
unxpdom	|- ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))

Proof of Theorem unxpdom

Step	Hyp	Ref	Expression
1		sdomex 4453	. . . 4 |- (1o ~< A -> (1o e. V /\ A e. V))
2	1	pm3.27d 325	. . 3 |- (1o ~< A -> A e. V)
3		sdomex 4453	. . . 4 |- (1o ~< B -> (1o e. V /\ B e. V))
4	3	pm3.27d 325	. . 3 |- (1o ~< B -> B e. V)
5	2, 4	anim12i 333	. 2 |- ((1o ~< A /\ 1o ~< B) -> (A e. V /\ B e. V))
6		breq2 2613	. . . . 5 |- (x = A -> (1o ~< x <-> 1o ~< A))
7	6	anbi1d 615	. . . 4 |- (x = A -> ((1o ~< x /\ 1o ~< y) <-> (1o ~< A /\ 1o ~< y)))
8		uneq1 2167	. . . . 5 |- (x = A -> (x u. y) = (A u. y))
9		xpeq1 3190	. . . . 5 |- (x = A -> (x X. y) = (A X. y))
10	8, 9	breq12d 2621	. . . 4 |- (x = A -> ((x u. y) ~<_ (x X. y) <-> (A u. y) ~<_ (A X. y)))
11	7, 10	imbi12d 624	. . 3 |- (x = A -> (((1o ~< x /\ 1o ~< y) -> (x u. y) ~<_ (x X. y)) <-> ((1o ~< A /\ 1o ~< y) -> (A u. y) ~<_ (A X. y))))
12		breq2 2613	. . . . 5 |- (y = B -> (1o ~< y <-> 1o ~< B))
13	12	anbi2d 614	. . . 4 |- (y = B -> ((1o ~< A /\ 1o ~< y) <-> (1o ~< A /\ 1o ~< B)))
14		uneq2 2168	. . . . 5 |- (y = B -> (A u. y) = (A u. B))
15		xpeq2 3191	. . . . 5 |- (y = B -> (A X. y) = (A X. B))
16	14, 15	breq12d 2621	. . . 4 |- (y = B -> ((A u. y) ~<_ (A X. y) <-> (A u. B) ~<_ (A X. B)))
17	13, 16	imbi12d 624	. . 3 |- (y = B -> (((1o ~< A /\ 1o ~< y) -> (A u. y) ~<_ (A X. y)) <-> ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))))
18		visset 1804	. . . 4 |- x e. V
19		visset 1804	. . . 4 |- y e. V
20	18, 19	unxpdomlem 4815	. . 3 |- ((1o ~< x /\ 1o ~< y) -> (x u. y) ~<_ (x X. y))
21	11, 17, 20	vtocl2g 1841	. 2 |- ((A e. V /\ B e. V) -> ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B)))
22	5, 21	mpcom 49	1 |- ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802   u. cun 2035   class class class wbr 2609   X. cxp 3158  1oc1o 4112   ~<_ cdom 4349   ~< csdm 4350

This theorem is referenced by:  unxpdom2 4817  sucxpdom 4818  infxpidmlem1 7495

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-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  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-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  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-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-2o 4118  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788

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