Theorem snon0 6930

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 4538	. . 3 |- -. A e. A
2		snidg 3621	. . . . . . 7 |- ( A e. V -> A e. { A } )
3	2	adantr 276	. . . . . 6 |- ( ( A e. V /\ { A } e. On ) -> A e. { A } )
4		ontr1 4387	. . . . . . 7 |- ( { A } e. On -> ( ( x e. A /\ A e. { A } ) -> x e. { A } ) )
5	4	adantl 277	. . . . . 6 |- ( ( A e. V /\ { A } e. On ) -> ( ( x e. A /\ A e. { A } ) -> x e. { A } ) )
6	3, 5	mpan2d 428	. . . . 5 |- ( ( A e. V /\ { A } e. On ) -> ( x e. A -> x e. { A } ) )
7		elsni 3610	. . . . 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 2240	. . . . 5 |- ( x = A -> ( x e. A <-> A e. A ) )
10	9	biimpcd 159	. . . 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 664	. 2 |- ( ( A e. V /\ { A } e. On ) -> -. x e. A )
13	12	eq0rdv 3467	1 |- ( ( A e. V /\ { A } e. On ) -> A = (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   (/)c0 3422   {csn 3592   Oncon0 4361

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4534

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-uni 3809  df-tr 4100  df-iord 4364  df-on 4366

This theorem is referenced by: (None)