Theorem dfac5lem2 8517

Description: Lemma for dfac5 8521. (Contributed by NM, 12-Apr-2004.)

Hypothesis

Ref	Expression
dfac5lem.1	|- A = { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) }

Assertion

Ref	Expression
dfac5lem2	|- ( <. w , g >. e. U. A <-> ( w e. h /\ g e. w ) )

Distinct variable groups:   w, u, t, h, g   w, A, g

Allowed substitution hints:   A( u, t, h)

Proof of Theorem dfac5lem2

Step	Hyp	Ref	Expression
1		dfac5lem.1	. . . 4 |- A = { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) }
2	1	unieqi 4260	. . 3 |- U. A = U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) }
3	2	eleq2i 2545	. 2 |- ( <. w , g >. e. U. A <-> <. w , g >. e. U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) } )
4		eluniab 4262	. . 3 |- ( <. w , g >. e. U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) } <-> E. u ( <. w , g >. e. u /\ ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) ) )
5		r19.42v 3021	. . . . 5 |- ( E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> ( ( <. w , g >. e. u /\ u =/= (/) ) /\ E. t e. h u = ( { t } X. t ) ) )
6		anass 649	. . . . 5 |- ( ( ( <. w , g >. e. u /\ u =/= (/) ) /\ E. t e. h u = ( { t } X. t ) ) <-> ( <. w , g >. e. u /\ ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) ) )
7	5, 6	bitr2i 250	. . . 4 |- ( ( <. w , g >. e. u /\ ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) ) <-> E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) )
8	7	exbii 1644	. . 3 |- ( E. u ( <. w , g >. e. u /\ ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) ) <-> E. u E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) )
9		rexcom4 3138	. . . 4 |- ( E. t e. h E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. u E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) )
10		df-rex 2823	. . . 4 |- ( E. t e. h E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) )
11	9, 10	bitr3i 251	. . 3 |- ( E. u E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) )
12	4, 8, 11	3bitri 271	. 2 |- ( <. w , g >. e. U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) } <-> E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) )
13		ancom 450	. . . . . . . . 9 |- ( ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> ( u = ( { t } X. t ) /\ ( <. w , g >. e. u /\ u =/= (/) ) ) )
14		ne0i 3796	. . . . . . . . . . 11 |- ( <. w , g >. e. u -> u =/= (/) )
15	14	pm4.71i 632	. . . . . . . . . 10 |- ( <. w , g >. e. u <-> ( <. w , g >. e. u /\ u =/= (/) ) )
16	15	anbi2i 694	. . . . . . . . 9 |- ( ( u = ( { t } X. t ) /\ <. w , g >. e. u ) <-> ( u = ( { t } X. t ) /\ ( <. w , g >. e. u /\ u =/= (/) ) ) )
17	13, 16	bitr4i 252	. . . . . . . 8 |- ( ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> ( u = ( { t } X. t ) /\ <. w , g >. e. u ) )
18	17	exbii 1644	. . . . . . 7 |- ( E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. u ( u = ( { t } X. t ) /\ <. w , g >. e. u ) )
19		snex 4694	. . . . . . . . 9 |- { t } e. _V
20		vex 3121	. . . . . . . . 9 |- t e. _V
21	19, 20	xpex 6599	. . . . . . . 8 |- ( { t } X. t ) e. _V
22		eleq2 2540	. . . . . . . 8 |- ( u = ( { t } X. t ) -> ( <. w , g >. e. u <-> <. w , g >. e. ( { t } X. t ) ) )
23	21, 22	ceqsexv 3155	. . . . . . 7 |- ( E. u ( u = ( { t } X. t ) /\ <. w , g >. e. u ) <-> <. w , g >. e. ( { t } X. t ) )
24	18, 23	bitri 249	. . . . . 6 |- ( E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> <. w , g >. e. ( { t } X. t ) )
25	24	anbi2i 694	. . . . 5 |- ( ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) <-> ( t e. h /\ <. w , g >. e. ( { t } X. t ) ) )
26		opelxp 5035	. . . . . . 7 |- ( <. w , g >. e. ( { t } X. t ) <-> ( w e. { t } /\ g e. t ) )
27		elsn 4047	. . . . . . . . 9 |- ( w e. { t } <-> w = t )
28		equcom 1743	. . . . . . . . 9 |- ( w = t <-> t = w )
29	27, 28	bitri 249	. . . . . . . 8 |- ( w e. { t } <-> t = w )
30	29	anbi1i 695	. . . . . . 7 |- ( ( w e. { t } /\ g e. t ) <-> ( t = w /\ g e. t ) )
31	26, 30	bitri 249	. . . . . 6 |- ( <. w , g >. e. ( { t } X. t ) <-> ( t = w /\ g e. t ) )
32	31	anbi2i 694	. . . . 5 |- ( ( t e. h /\ <. w , g >. e. ( { t } X. t ) ) <-> ( t e. h /\ ( t = w /\ g e. t ) ) )
33		an12 795	. . . . 5 |- ( ( t e. h /\ ( t = w /\ g e. t ) ) <-> ( t = w /\ ( t e. h /\ g e. t ) ) )
34	25, 32, 33	3bitri 271	. . . 4 |- ( ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) <-> ( t = w /\ ( t e. h /\ g e. t ) ) )
35	34	exbii 1644	. . 3 |- ( E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) <-> E. t ( t = w /\ ( t e. h /\ g e. t ) ) )
36		vex 3121	. . . 4 |- w e. _V
37		elequ1 1770	. . . . 5 |- ( t = w -> ( t e. h <-> w e. h ) )
38		eleq2 2540	. . . . 5 |- ( t = w -> ( g e. t <-> g e. w ) )
39	37, 38	anbi12d 710	. . . 4 |- ( t = w -> ( ( t e. h /\ g e. t ) <-> ( w e. h /\ g e. w ) ) )
40	36, 39	ceqsexv 3155	. . 3 |- ( E. t ( t = w /\ ( t e. h /\ g e. t ) ) <-> ( w e. h /\ g e. w ) )
41	35, 40	bitri 249	. 2 |- ( E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) <-> ( w e. h /\ g e. w ) )
42	3, 12, 41	3bitri 271	1 |- ( <. w , g >. e. U. A <-> ( w e. h /\ g e. w ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   =/= wne 2662   E.wrex 2818   (/)c0 3790   {csn 4033   <.cop 4039   U.cuni 4251   X. cxp 5003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-opab 4512  df-xp 5011

This theorem is referenced by:  dfac5lem5  8520