Theorem 2omotaplemst 7614

Description: Lemma for 2omotap 7615. (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 7612	. . . 4 |- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }
2		2omotaplemap 7613	. . . . . 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 6784	. . . . . . . . . 10 |- 2o e. om
5	4	elexi 2834	. . . . . . . . 9 |- 2o e. _V
6	5, 5	xpex 4886	. . . . . . . 8 |- ( 2o X. 2o ) e. _V
7		opabssxp 4844	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } C_ ( 2o X. 2o )
8	6, 7	ssexi 4266	. . . . . . 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 4844	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } C_ ( 2o X. 2o )
11	6, 10	ssexi 4266	. . . . . . 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 7611	. . . . . . 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 7608	. . . . . . 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 7608	. . . . . . 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 3008	. . . . . 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 1284	. . . . 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 2327	. . 3 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
22		0lt2o 6704	. . . 4 |- (/) e. 2o
23		1lt2o 6705	. . . 4 |- 1o e. 2o
24		neeq1 2433	. . . . . 6 |- ( u = (/) -> ( u =/= v <-> (/) =/= v ) )
25	24	anbi2d 468	. . . . 5 |- ( u = (/) -> ( ( ph /\ u =/= v ) <-> ( ph /\ (/) =/= v ) ) )
26		neeq2 2434	. . . . . 6 |- ( v = 1o -> ( (/) =/= v <-> (/) =/= 1o ) )
27	26	anbi2d 468	. . . . 5 |- ( v = 1o -> ( ( ph /\ (/) =/= v ) <-> ( ph /\ (/) =/= 1o ) ) )
28	25, 27	opelopab2 4408	. . . 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 430	. . 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 1402   E*wmo 2087   e. wcel 2209   =/= wne 2420   _Vcvv 2821   (/)c0 3520   <.cop 3708   {copab 4186   omcom 4732   X. cxp 4767   1oc1o 6670   2oc2o 6671   TAp wtap 7604

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-1o 6677  df-2o 6678  df-pap 7598  df-tap 7605

This theorem is referenced by:  2omotap  7615