Theorem 2omotaplemst 7476

Description: Lemma for 2omotap 7477. (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 7474	. . . 4 |- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }
2		2omotaplemap 7475	. . . . . 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 6688	. . . . . . . . . 10 |- 2o e. om
5	4	elexi 2815	. . . . . . . . 9 |- 2o e. _V
6	5, 5	xpex 4842	. . . . . . . 8 |- ( 2o X. 2o ) e. _V
7		opabssxp 4800	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } C_ ( 2o X. 2o )
8	6, 7	ssexi 4227	. . . . . . 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 4800	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } C_ ( 2o X. 2o )
11	6, 10	ssexi 4227	. . . . . . 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 7473	. . . . . . 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 7470	. . . . . . 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 7470	. . . . . . 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 2988	. . . . . 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 1279	. . . . 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 2320	. . 3 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
22		0lt2o 6608	. . . 4 |- (/) e. 2o
23		1lt2o 6609	. . . 4 |- 1o e. 2o
24		neeq1 2415	. . . . . 6 |- ( u = (/) -> ( u =/= v <-> (/) =/= v ) )
25	24	anbi2d 464	. . . . 5 |- ( u = (/) -> ( ( ph /\ u =/= v ) <-> ( ph /\ (/) =/= v ) ) )
26		neeq2 2416	. . . . . 6 |- ( v = 1o -> ( (/) =/= v <-> (/) =/= 1o ) )
27	26	anbi2d 464	. . . . 5 |- ( v = 1o -> ( ( ph /\ (/) =/= v ) <-> ( ph /\ (/) =/= 1o ) ) )
28	25, 27	opelopab2 4365	. . . 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 1397   E*wmo 2080   e. wcel 2202   =/= wne 2402   _Vcvv 2802   (/)c0 3494   <.cop 3672   {copab 4149   omcom 4688   X. cxp 4723   1oc1o 6574   2oc2o 6575   TAp wtap 7467

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-1o 6581  df-2o 6582  df-pap 7466  df-tap 7468

This theorem is referenced by:  2omotap  7477