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Theorem ssexnelpss 3892
Description: If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020.)
Assertion
Ref Expression
ssexnelpss
Distinct variable groups:   ,   ,

Proof of Theorem ssexnelpss
StepHypRef Expression
1 df-nel 2655 . . . 4
2 ssnelpss 3890 . . . . 5
32expdimp 437 . . . 4
41, 3syl5bi 217 . . 3
54rexlimdva 2949 . 2
65imp 429 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  e.wcel 1818  e/wnel 2653  E.wrex 2808  C_wss 3475  C.wpss 3476
This theorem is referenced by:  sgrpssmgm  16051  mndsssgrp  16052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-cleq 2449  df-clel 2452  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-pss 3491
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