Theorem snon0 7133

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 4639	. . 3 |- -. A e. A
2		snidg 3698	. . . . . . 7 |- ( A e. V -> A e. { A } )
3	2	adantr 276	. . . . . 6 |- ( ( A e. V /\ { A } e. On ) -> A e. { A } )
4		ontr1 4486	. . . . . . 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 3687	. . . . 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 2294	. . . . 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 3539	1 |- ( ( A e. V /\ { A } e. On ) -> A = (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   (/)c0 3494   {csn 3669   Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by: (None)