Theorem unxpdom 8167

Description: Cartesian product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

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

Proof of Theorem unxpdom

Dummy variables x y u t v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsdom 7962	. . . 4 |- Rel ~<
2	1	brrelex2i 5159	. . 3 |- ( 1o ~< A -> A e. _V )
3	1	brrelex2i 5159	. . 3 |- ( 1o ~< B -> B e. _V )
4	2, 3	anim12i 590	. 2 |- ( ( 1o ~< A /\ 1o ~< B ) -> ( A e. _V /\ B e. _V ) )
5		breq2 4657	. . . . 5 |- ( x = A -> ( 1o ~< x <-> 1o ~< A ) )
6	5	anbi1d 741	. . . 4 |- ( x = A -> ( ( 1o ~< x /\ 1o ~< y ) <-> ( 1o ~< A /\ 1o ~< y ) ) )
7		uneq1 3760	. . . . 5 |- ( x = A -> ( x u. y ) = ( A u. y ) )
8		xpeq1 5128	. . . . 5 |- ( x = A -> ( x X. y ) = ( A X. y ) )
9	7, 8	breq12d 4666	. . . 4 |- ( x = A -> ( ( x u. y ) ~<_ ( x X. y ) <-> ( A u. y ) ~<_ ( A X. y ) ) )
10	6, 9	imbi12d 334	. . 3 |- ( x = A -> ( ( ( 1o ~< x /\ 1o ~< y ) -> ( x u. y ) ~<_ ( x X. y ) ) <-> ( ( 1o ~< A /\ 1o ~< y ) -> ( A u. y ) ~<_ ( A X. y ) ) ) )
11		breq2 4657	. . . . 5 |- ( y = B -> ( 1o ~< y <-> 1o ~< B ) )
12	11	anbi2d 740	. . . 4 |- ( y = B -> ( ( 1o ~< A /\ 1o ~< y ) <-> ( 1o ~< A /\ 1o ~< B ) ) )
13		uneq2 3761	. . . . 5 |- ( y = B -> ( A u. y ) = ( A u. B ) )
14		xpeq2 5129	. . . . 5 |- ( y = B -> ( A X. y ) = ( A X. B ) )
15	13, 14	breq12d 4666	. . . 4 |- ( y = B -> ( ( A u. y ) ~<_ ( A X. y ) <-> ( A u. B ) ~<_ ( A X. B ) ) )
16	12, 15	imbi12d 334	. . 3 |- ( y = B -> ( ( ( 1o ~< A /\ 1o ~< y ) -> ( A u. y ) ~<_ ( A X. y ) ) <-> ( ( 1o ~< A /\ 1o ~< B ) -> ( A u. B ) ~<_ ( A X. B ) ) ) )
17		eqid 2622	. . . 4 |- ( z e. ( x u. y ) |-> if ( z e. x , <. z , if ( z = v , w , t ) >. , <. if ( z = w , u , v ) , z >. ) ) = ( z e. ( x u. y ) |-> if ( z e. x , <. z , if ( z = v , w , t ) >. , <. if ( z = w , u , v ) , z >. ) )
18		eqid 2622	. . . 4 |- if ( z e. x , <. z , if ( z = v , w , t ) >. , <. if ( z = w , u , v ) , z >. ) = if ( z e. x , <. z , if ( z = v , w , t ) >. , <. if ( z = w , u , v ) , z >. )
19	17, 18	unxpdomlem3 8166	. . 3 |- ( ( 1o ~< x /\ 1o ~< y ) -> ( x u. y ) ~<_ ( x X. y ) )
20	10, 16, 19	vtocl2g 3270	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( ( 1o ~< A /\ 1o ~< B ) -> ( A u. B ) ~<_ ( A X. B ) ) )
21	4, 20	mpcom 38	1 |- ( ( 1o ~< A /\ 1o ~< B ) -> ( A u. B ) ~<_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   ifcif 4086   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   1oc1o 7553   ~<_ cdom 7953   ~< csdm 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  unxpdom2  8168  sucxpdom  8169  cdaxpdom  9011