Theorem 2omotaplemst 7370

Description: Lemma for 2omotap 7371. (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 7368	. . . 4 |- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }
2		2omotaplemap 7369	. . . . . 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 6607	. . . . . . . . . 10 |- 2o e. om
5	4	elexi 2784	. . . . . . . . 9 |- 2o e. _V
6	5, 5	xpex 4790	. . . . . . . 8 |- ( 2o X. 2o ) e. _V
7		opabssxp 4749	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } C_ ( 2o X. 2o )
8	6, 7	ssexi 4182	. . . . . . 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 4749	. . . . . . . 8 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } C_ ( 2o X. 2o )
11	6, 10	ssexi 4182	. . . . . . 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 7367	. . . . . . 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 7364	. . . . . . 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 7364	. . . . . . 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 2955	. . . . . 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 1256	. . . . 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 2294	. . 3 |- ( ( E* r r TAp 2o /\ -. -. ph ) -> <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
22		0lt2o 6527	. . . 4 |- (/) e. 2o
23		1lt2o 6528	. . . 4 |- 1o e. 2o
24		neeq1 2389	. . . . . 6 |- ( u = (/) -> ( u =/= v <-> (/) =/= v ) )
25	24	anbi2d 464	. . . . 5 |- ( u = (/) -> ( ( ph /\ u =/= v ) <-> ( ph /\ (/) =/= v ) ) )
26		neeq2 2390	. . . . . 6 |- ( v = 1o -> ( (/) =/= v <-> (/) =/= 1o ) )
27	26	anbi2d 464	. . . . 5 |- ( v = 1o -> ( ( ph /\ (/) =/= v ) <-> ( ph /\ (/) =/= 1o ) ) )
28	25, 27	opelopab2 4317	. . . 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 1373   E*wmo 2055   e. wcel 2176   =/= wne 2376   _Vcvv 2772   (/)c0 3460   <.cop 3636   {copab 4104   omcom 4638   X. cxp 4673   1oc1o 6495   2oc2o 6496   TAp wtap 7361

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-1o 6502  df-2o 6503  df-pap 7360  df-tap 7362

This theorem is referenced by:  2omotap  7371