Theorem 2oneel 7586

Description: (/) and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.)

Assertion

Ref	Expression
2oneel	|- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }

Distinct variable group:   v, u

Proof of Theorem 2oneel

Step	Hyp	Ref	Expression
1		1n0 6678	. . 3 |- 1o =/= (/)
2	1	necomi 2499	. 2 |- (/) =/= 1o
3		0lt2o 6687	. . 3 |- (/) e. 2o
4		1lt2o 6688	. . 3 |- 1o e. 2o
5		neeq1 2427	. . . 4 |- ( u = (/) -> ( u =/= v <-> (/) =/= v ) )
6		neeq2 2428	. . . 4 |- ( v = 1o -> ( (/) =/= v <-> (/) =/= 1o ) )
7	5, 6	opelopab2 4394	. . 3 |- ( ( (/) e. 2o /\ 1o e. 2o ) -> ( <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } <-> (/) =/= 1o ) )
8	3, 4, 7	mp2an 426	. 2 |- ( <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } <-> (/) =/= 1o )
9	2, 8	mpbir 146	1 |- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2205   =/= wne 2414   (/)c0 3512   <.cop 3697   {copab 4175   1oc1o 6653   2oc2o 6654

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-opab 4177  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-1o 6660  df-2o 6661

This theorem is referenced by:  2omotaplemst  7588