Theorem dfac5lem2 8560

Description: Lemma for dfac5 8564. (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 4210	. . 3 |- U. A = U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) }
3	2	eleq2i 2523	. 2 |- ( <. w , g >. e. U. A <-> <. w , g >. e. U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) } )
4		eluniab 4212	. . 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 2947	. . . . 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 655	. . . . 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 254	. . . 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 1720	. . 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 3069	. . . 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 2745	. . . 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 255	. . 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 275	. 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 452	. . . . . . . . 9 |- ( ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> ( u = ( { t } X. t ) /\ ( <. w , g >. e. u /\ u =/= (/) ) ) )
14		ne0i 3739	. . . . . . . . . . 11 |- ( <. w , g >. e. u -> u =/= (/) )
15	14	pm4.71i 638	. . . . . . . . . 10 |- ( <. w , g >. e. u <-> ( <. w , g >. e. u /\ u =/= (/) ) )
16	15	anbi2i 701	. . . . . . . . 9 |- ( ( u = ( { t } X. t ) /\ <. w , g >. e. u ) <-> ( u = ( { t } X. t ) /\ ( <. w , g >. e. u /\ u =/= (/) ) ) )
17	13, 16	bitr4i 256	. . . . . . . 8 |- ( ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> ( u = ( { t } X. t ) /\ <. w , g >. e. u ) )
18	17	exbii 1720	. . . . . . 7 |- ( E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. u ( u = ( { t } X. t ) /\ <. w , g >. e. u ) )
19		snex 4644	. . . . . . . . 9 |- { t } e. _V
20		vex 3050	. . . . . . . . 9 |- t e. _V
21	19, 20	xpex 6600	. . . . . . . 8 |- ( { t } X. t ) e. _V
22		eleq2 2520	. . . . . . . 8 |- ( u = ( { t } X. t ) -> ( <. w , g >. e. u <-> <. w , g >. e. ( { t } X. t ) ) )
23	21, 22	ceqsexv 3086	. . . . . . 7 |- ( E. u ( u = ( { t } X. t ) /\ <. w , g >. e. u ) <-> <. w , g >. e. ( { t } X. t ) )
24	18, 23	bitri 253	. . . . . 6 |- ( E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> <. w , g >. e. ( { t } X. t ) )
25	24	anbi2i 701	. . . . 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 4867	. . . . . . 7 |- ( <. w , g >. e. ( { t } X. t ) <-> ( w e. { t } /\ g e. t ) )
27		elsn 3984	. . . . . . . . 9 |- ( w e. { t } <-> w = t )
28		equcom 1864	. . . . . . . . 9 |- ( w = t <-> t = w )
29	27, 28	bitri 253	. . . . . . . 8 |- ( w e. { t } <-> t = w )
30	29	anbi1i 702	. . . . . . 7 |- ( ( w e. { t } /\ g e. t ) <-> ( t = w /\ g e. t ) )
31	26, 30	bitri 253	. . . . . 6 |- ( <. w , g >. e. ( { t } X. t ) <-> ( t = w /\ g e. t ) )
32	31	anbi2i 701	. . . . 5 |- ( ( t e. h /\ <. w , g >. e. ( { t } X. t ) ) <-> ( t e. h /\ ( t = w /\ g e. t ) ) )
33		an12 807	. . . . 5 |- ( ( t e. h /\ ( t = w /\ g e. t ) ) <-> ( t = w /\ ( t e. h /\ g e. t ) ) )
34	25, 32, 33	3bitri 275	. . . 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 1720	. . 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 3050	. . . 4 |- w e. _V
37		elequ1 1896	. . . . 5 |- ( t = w -> ( t e. h <-> w e. h ) )
38		eleq2 2520	. . . . 5 |- ( t = w -> ( g e. t <-> g e. w ) )
39	37, 38	anbi12d 718	. . . 4 |- ( t = w -> ( ( t e. h /\ g e. t ) <-> ( w e. h /\ g e. w ) ) )
40	36, 39	ceqsexv 3086	. . 3 |- ( E. t ( t = w /\ ( t e. h /\ g e. t ) ) <-> ( w e. h /\ g e. w ) )
41	35, 40	bitri 253	. 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 275	1 |- ( <. w , g >. e. U. A <-> ( w e. h /\ g e. w ) )

Syntax hints:   <-> wb 188   /\ wa 371   = wceq 1446   E.wex 1665   e. wcel 1889   {cab 2439   =/= wne 2624   E.wrex 2740   (/)c0 3733   {csn 3970   <.cop 3976   U.cuni 4201   X. cxp 4835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-opab 4465  df-xp 4843  df-rel 4844

This theorem is referenced by:  dfac5lem5  8563