Theorem ssnel 4711

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 4683	. 2 |- -. B e. B
2		ssel 3242	. 2 |- ( A C_ B -> ( B e. A -> B e. B ) )
3	1, 2	mtoi 674	1 |- ( A C_ B -> -. B e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2209   C_ wss 3220

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-sn 3711

This theorem is referenced by:  nntri1  6759  pw1ne3  7579  3nelsucpw1  7583  3nsssucpw1  7585  nninfctlemfo  12795