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Theorem ssnel 4564
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
StepHypRef Expression
1 elirr 4536 . 2  |-  -.  B  e.  B
2 ssel 3149 . 2  |-  ( A 
C_  B  ->  ( B  e.  A  ->  B  e.  B ) )
31, 2mtoi 664 1  |-  ( A 
C_  B  ->  -.  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 2148    C_ wss 3129
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-sn 3597
This theorem is referenced by:  nntri1  6490  pw1ne3  7222  3nelsucpw1  7226  3nsssucpw1  7228
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