Theorem snon0 6909

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 4523	. . 3 |- -. A e. A
2		snidg 3610	. . . . . . 7 |- ( A e. V -> A e. { A } )
3	2	adantr 274	. . . . . 6 |- ( ( A e. V /\ { A } e. On ) -> A e. { A } )
4		ontr1 4372	. . . . . . 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 426	. . . . 5 |- ( ( A e. V /\ { A } e. On ) -> ( x e. A -> x e. { A } ) )
7		elsni 3599	. . . . 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 2233	. . . . 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 659	. 2 |- ( ( A e. V /\ { A } e. On ) -> -. x e. A )
13	12	eq0rdv 3458	1 |- ( ( A e. V /\ { A } e. On ) -> A = (/) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   (/)c0 3414   {csn 3581   Oncon0 4346

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4519

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3587  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351

This theorem is referenced by: (None)