Theorem ssnel 4667

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 4639	. 2 |- -. B e. B
2		ssel 3221	. 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 2202   C_ wss 3200

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-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-sn 3675

This theorem is referenced by:  nntri1  6663  pw1ne3  7447  3nelsucpw1  7451  3nsssucpw1  7453  nninfctlemfo  12610