Theorem 2omotaplemst 7257

Description: Lemma for 2omotap 7258. (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 7255	. . . 4 |- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }
2		2omotaplemap 7256	. . . . . 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 6522	. . . . . . . . . 10 |- 2o e. om
5	4	elexi 2750	. . . . . . . . 9 |- 2o e. _V
6	5, 5	xpex 4742	. . . . . . . 8 |- ( 2o X. 2o ) e. _V
7		opabssxp 4701	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } C_ ( 2o X. 2o )
8	6, 7	ssexi 4142	. . . . . . 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 4701	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } C_ ( 2o X. 2o )
11	6, 10	ssexi 4142	. . . . . . 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 7254	. . . . . . 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 7251	. . . . . . 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 7251	. . . . . . 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 2920	. . . . . 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 1244	. . . . 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 2266	. . 3 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
22		0lt2o 6442	. . . 4 |- (/) e. 2o
23		1lt2o 6443	. . . 4 |- 1o e. 2o
24		neeq1 2360	. . . . . 6 |- ( u = (/) -> ( u =/= v <-> (/) =/= v ) )
25	24	anbi2d 464	. . . . 5 |- ( u = (/) -> ( ( ph /\ u =/= v ) <-> ( ph /\ (/) =/= v ) ) )
26		neeq2 2361	. . . . . 6 |- ( v = 1o -> ( (/) =/= v <-> (/) =/= 1o ) )
27	26	anbi2d 464	. . . . 5 |- ( v = 1o -> ( ( ph /\ (/) =/= v ) <-> ( ph /\ (/) =/= 1o ) ) )
28	25, 27	opelopab2 4271	. . . 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 1353   E*wmo 2027   e. wcel 2148   =/= wne 2347   _Vcvv 2738   (/)c0 3423   <.cop 3596   {copab 4064   omcom 4590   X. cxp 4625   1oc1o 6410   2oc2o 6411   TAp wtap 7248

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-1o 6417  df-2o 6418  df-pap 7247  df-tap 7249

This theorem is referenced by:  2omotap  7258