Theorem disjxp1 39238

Description: The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypothesis

Ref	Expression
disjxp1.1	|- ( ph -> Disj x e. A B )

Assertion

Ref	Expression
disjxp1	|- ( ph -> Disj x e. A ( B X. C ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)   C( x)

Proof of Theorem disjxp1

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 400	. . . . 5 |- ( y = z -> ( y = z \/ ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) ) )
2	1	adantl 482	. . . 4 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y = z ) -> ( y = z \/ ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) ) )
3		simpl 473	. . . . 5 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ -. y = z ) -> ( ph /\ ( y e. A /\ z e. A ) ) )
4		neqne 2802	. . . . . 6 |- ( -. y = z -> y =/= z )
5	4	adantl 482	. . . . 5 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ -. y = z ) -> y =/= z )
6		csbxp 5200	. . . . . . . . 9 |- [_ y / x ]_ ( B X. C ) = ( [_ y / x ]_ B X. [_ y / x ]_ C )
7		csbxp 5200	. . . . . . . . 9 |- [_ z / x ]_ ( B X. C ) = ( [_ z / x ]_ B X. [_ z / x ]_ C )
8	6, 7	ineq12i 3812	. . . . . . . 8 |- ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = ( ( [_ y / x ]_ B X. [_ y / x ]_ C ) i^i ( [_ z / x ]_ B X. [_ z / x ]_ C ) )
9	8	a1i 11	. . . . . . 7 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = ( ( [_ y / x ]_ B X. [_ y / x ]_ C ) i^i ( [_ z / x ]_ B X. [_ z / x ]_ C ) ) )
10		simpll 790	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> ph )
11		simplrl 800	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> y e. A )
12		simplrr 801	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> z e. A )
13	10, 11, 12	jca31 557	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> ( ( ph /\ y e. A ) /\ z e. A ) )
14		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> y =/= z )
15	14	neneqd 2799	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> -. y = z )
16		disjxp1.1	. . . . . . . . . . . . 13 |- ( ph -> Disj x e. A B )
17		disjors 4635	. . . . . . . . . . . . 13 |- (Disj x e. A B <-> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
18	16, 17	sylib 208	. . . . . . . . . . . 12 |- ( ph -> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
19	18	r19.21bi 2932	. . . . . . . . . . 11 |- ( ( ph /\ y e. A ) -> A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
20	19	r19.21bi 2932	. . . . . . . . . 10 |- ( ( ( ph /\ y e. A ) /\ z e. A ) -> ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
21	20	ord 392	. . . . . . . . 9 |- ( ( ( ph /\ y e. A ) /\ z e. A ) -> ( -. y = z -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
22	13, 15, 21	sylc 65	. . . . . . . 8 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) )
23		xpdisj1 5555	. . . . . . . 8 |- ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) -> ( ( [_ y / x ]_ B X. [_ y / x ]_ C ) i^i ( [_ z / x ]_ B X. [_ z / x ]_ C ) ) = (/) )
24	22, 23	syl 17	. . . . . . 7 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> ( ( [_ y / x ]_ B X. [_ y / x ]_ C ) i^i ( [_ z / x ]_ B X. [_ z / x ]_ C ) ) = (/) )
25	9, 24	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) )
26		olc 399	. . . . . 6 |- ( ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) -> ( y = z \/ ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) ) )
27	25, 26	syl 17	. . . . 5 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ y =/= z ) -> ( y = z \/ ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) ) )
28	3, 5, 27	syl2anc 693	. . . 4 |- ( ( ( ph /\ ( y e. A /\ z e. A ) ) /\ -. y = z ) -> ( y = z \/ ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) ) )
29	2, 28	pm2.61dan 832	. . 3 |- ( ( ph /\ ( y e. A /\ z e. A ) ) -> ( y = z \/ ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) ) )
30	29	ralrimivva 2971	. 2 |- ( ph -> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) ) )
31		disjors 4635	. 2 |- (Disj x e. A ( B X. C ) <-> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ ( B X. C ) i^i [_ z / x ]_ ( B X. C ) ) = (/) ) )
32	30, 31	sylibr 224	1 |- ( ph -> Disj x e. A ( B X. C ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   i^i cin 3573   (/)c0 3915  Disj wdisj 4620   X. cxp 5112

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-disj 4621  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  disjsnxp  39239