Theorem 2oneel 7442

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 6578	. . 3 |- 1o =/= (/)
2	1	necomi 2485	. 2 |- (/) =/= 1o
3		0lt2o 6587	. . 3 |- (/) e. 2o
4		1lt2o 6588	. . 3 |- 1o e. 2o
5		neeq1 2413	. . . 4 |- ( u = (/) -> ( u =/= v <-> (/) =/= v ) )
6		neeq2 2414	. . . 4 |- ( v = 1o -> ( (/) =/= v <-> (/) =/= 1o ) )
7	5, 6	opelopab2 4359	. . 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 2200   =/= wne 2400   (/)c0 3491   <.cop 3669   {copab 4144   1oc1o 6555   2oc2o 6556

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-opab 4146  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462  df-1o 6562  df-2o 6563

This theorem is referenced by:  2omotaplemst  7444