Theorem unxpdom 4844

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 4473	. . . 4 |- (1o ~< A -> (1o e. V /\ A e. V))
2	1	pm3.27d 325	. . 3 |- (1o ~< A -> A e. V)
3		sdomex 4473	. . . 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 2623	. . . . 5 |- (x = A -> (1o ~< x <-> 1o ~< A))
7	6	anbi1d 617	. . . 4 |- (x = A -> ((1o ~< x /\ 1o ~< y) <-> (1o ~< A /\ 1o ~< y)))
8		uneq1 2177	. . . . 5 |- (x = A -> (x u. y) = (A u. y))
9		xpeq1 3200	. . . . 5 |- (x = A -> (x X. y) = (A X. y))
10	8, 9	breq12d 2631	. . . 4 |- (x = A -> ((x u. y) ~<_ (x X. y) <-> (A u. y) ~<_ (A X. y)))
11	7, 10	imbi12d 626	. . 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 2623	. . . . 5 |- (y = B -> (1o ~< y <-> 1o ~< B))
13	12	anbi2d 616	. . . 4 |- (y = B -> ((1o ~< A /\ 1o ~< y) <-> (1o ~< A /\ 1o ~< B)))
14		uneq2 2178	. . . . 5 |- (y = B -> (A u. y) = (A u. B))
15		xpeq2 3201	. . . . 5 |- (y = B -> (A X. y) = (A X. B))
16	14, 15	breq12d 2631	. . . 4 |- (y = B -> ((A u. y) ~<_ (A X. y) <-> (A u. B) ~<_ (A X. B)))
17	13, 16	imbi12d 626	. . 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 1813	. . . 4 |- x e. V
19		visset 1813	. . . 4 |- y e. V
20	18, 19	unxpdomlem 4843	. . 3 |- ((1o ~< x /\ 1o ~< y) -> (x u. y) ~<_ (x X. y))
21	11, 17, 20	vtocl2g 1850	. 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 956   e. wcel 958  Vcvv 1811   u. cun 2045   class class class wbr 2619   X. cxp 3168  1oc1o 4128   ~<_ cdom 4365   ~< csdm 4366

This theorem is referenced by:  unxpdom2 4845  sucxpdom 4846  infxpidmlem1 7552

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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  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-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-2o 4134  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816

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