Theorem 2omotaplemst 7444

Description: Lemma for 2omotap 7445. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotaplemst	|- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )

Distinct variable group:   ph, r

Proof of Theorem 2omotaplemst

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2oneel 7442	. . . 4 |- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }
2		2omotaplemap 7443	. . . . . 6 |- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )
3	2	adantl 277	. . . . 5 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )
4		2onn 6667	. . . . . . . . . 10 |- 2o e. om
5	4	elexi 2812	. . . . . . . . 9 |- 2o e. _V
6	5, 5	xpex 4834	. . . . . . . 8 |- ( 2o X. 2o ) e. _V
7		opabssxp 4793	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } C_ ( 2o X. 2o )
8	6, 7	ssexi 4222	. . . . . . 7 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } e. _V
9	8	a1i 9	. . . . . 6 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } e. _V )
10		opabssxp 4793	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } C_ ( 2o X. 2o )
11	6, 10	ssexi 4222	. . . . . . 7 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } e. _V
12	11	a1i 9	. . . . . 6 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } e. _V )
13		simpl 109	. . . . . 6 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> E* r r TAp 2o )
14		2onetap 7441	. . . . . . 7 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } TAp 2o
15	14	a1i 9	. . . . . 6 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } TAp 2o )
16		tapeq1 7438	. . . . . . 7 |- ( r = { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } -> ( r TAp 2o <-> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } TAp 2o ) )
17		tapeq1 7438	. . . . . . 7 |- ( r = { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } -> ( r TAp 2o <-> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o ) )
18	16, 17	mob 2985	. . . . . 6 |- ( ( ( { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } e. _V /\ { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } e. _V ) /\ E* r r TAp 2o /\ { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } TAp 2o ) -> ( { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } = { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o ) )
19	9, 12, 13, 15, 18	syl211anc 1277	. . . . 5 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> ( { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } = { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o ) )
20	3, 19	mpbird 167	. . . 4 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } = { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
21	1, 20	eleqtrid 2318	. . 3 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
22		0lt2o 6587	. . . 4 |- (/) e. 2o
23		1lt2o 6588	. . . 4 |- 1o e. 2o
24		neeq1 2413	. . . . . 6 |- ( u = (/) -> ( u =/= v <-> (/) =/= v ) )
25	24	anbi2d 464	. . . . 5 |- ( u = (/) -> ( ( ph /\ u =/= v ) <-> ( ph /\ (/) =/= v ) ) )
26		neeq2 2414	. . . . . 6 |- ( v = 1o -> ( (/) =/= v <-> (/) =/= 1o ) )
27	26	anbi2d 464	. . . . 5 |- ( v = 1o -> ( ( ph /\ (/) =/= v ) <-> ( ph /\ (/) =/= 1o ) ) )
28	25, 27	opelopab2 4359	. . . 4 |- ( ( (/) e. 2o /\ 1o e. 2o ) -> ( <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ (/) =/= 1o ) ) )
29	22, 23, 28	mp2an 426	. . 3 |- ( <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ (/) =/= 1o ) )
30	21, 29	sylib 122	. 2 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> ( ph /\ (/) =/= 1o ) )
31	30	simpld 112	1 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E*wmo 2078   e. wcel 2200   =/= wne 2400   _Vcvv 2799   (/)c0 3491   <.cop 3669   {copab 4144   omcom 4682   X. cxp 4717   1oc1o 6555   2oc2o 6556   TAp wtap 7435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-1o 6562  df-2o 6563  df-pap 7434  df-tap 7436

This theorem is referenced by:  2omotap  7445