Theorem snon0 6901

Description: An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)

Assertion

Ref	Expression
snon0	|- ( ( A e. V /\ { A } e. On ) -> A = (/) )

Proof of Theorem snon0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elirr 4518	. . 3 |- -. A e. A
2		snidg 3605	. . . . . . 7 |- ( A e. V -> A e. { A } )
3	2	adantr 274	. . . . . 6 |- ( ( A e. V /\ { A } e. On ) -> A e. { A } )
4		ontr1 4367	. . . . . . 7 |- ( { A } e. On -> ( ( x e. A /\ A e. { A } ) -> x e. { A } ) )
5	4	adantl 275	. . . . . 6 |- ( ( A e. V /\ { A } e. On ) -> ( ( x e. A /\ A e. { A } ) -> x e. { A } ) )
6	3, 5	mpan2d 425	. . . . 5 |- ( ( A e. V /\ { A } e. On ) -> ( x e. A -> x e. { A } ) )
7		elsni 3594	. . . . 5 |- ( x e. { A } -> x = A )
8	6, 7	syl6 33	. . . 4 |- ( ( A e. V /\ { A } e. On ) -> ( x e. A -> x = A ) )
9		eleq1 2229	. . . . 5 |- ( x = A -> ( x e. A <-> A e. A ) )
10	9	biimpcd 158	. . . 4 |- ( x e. A -> ( x = A -> A e. A ) )
11	8, 10	sylcom 28	. . 3 |- ( ( A e. V /\ { A } e. On ) -> ( x e. A -> A e. A ) )
12	1, 11	mtoi 654	. 2 |- ( ( A e. V /\ { A } e. On ) -> -. x e. A )
13	12	eq0rdv 3453	1 |- ( ( A e. V /\ { A } e. On ) -> A = (/) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   (/)c0 3409   {csn 3576   Oncon0 4341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by: (None)