Theorem ssnel 4691

Description: Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.)

Assertion

Ref	Expression
ssnel	|- ( A C_ B -> -. B e. A )

Proof of Theorem ssnel

Step	Hyp	Ref	Expression
1		elirr 4663	. 2 |- -. B e. B
2		ssel 3232	. 2 |- ( A C_ B -> ( B e. A -> B e. B ) )
3	1, 2	mtoi 670	1 |- ( A C_ B -> -. B e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2203   C_ wss 3211

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-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-sn 3695

This theorem is referenced by:  nntri1  6729  pw1ne3  7540  3nelsucpw1  7544  3nsssucpw1  7546  nninfctlemfo  12736