Theorem snon0 7202

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 4663	. . 3 |- -. A e. A
2		snidg 3718	. . . . . . 7 |- ( A e. V -> A e. { A } )
3	2	adantr 276	. . . . . 6 |- ( ( A e. V /\ { A } e. On ) -> A e. { A } )
4		ontr1 4510	. . . . . . 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 3707	. . . . 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 2295	. . . . 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 670	. 2 |- ( ( A e. V /\ { A } e. On ) -> -. x e. A )
13	12	eq0rdv 3553	1 |- ( ( A e. V /\ { A } e. On ) -> A = (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   (/)c0 3508   {csn 3689   Oncon0 4484

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-ext 2214  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489

This theorem is referenced by: (None)