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Theorem ssexnelpss 4112
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 ((𝐴𝐵 ∧ ∃𝑥𝐵 𝑥𝐴) → 𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssexnelpss
StepHypRef Expression
1 df-nel 3045 . . . 4 (𝑥𝐴 ↔ ¬ 𝑥𝐴)
2 ssnelpss 4110 . . . . 5 (𝐴𝐵 → ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → 𝐴𝐵))
32expdimp 451 . . . 4 ((𝐴𝐵𝑥𝐵) → (¬ 𝑥𝐴𝐴𝐵))
41, 3biimtrid 241 . . 3 ((𝐴𝐵𝑥𝐵) → (𝑥𝐴𝐴𝐵))
54rexlimdva 3153 . 2 (𝐴𝐵 → (∃𝑥𝐵 𝑥𝐴𝐴𝐵))
65imp 405 1 ((𝐴𝐵 ∧ ∃𝑥𝐵 𝑥𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wcel 2104  wnel 3044  wrex 3068  wss 3947  wpss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-cleq 2722  df-clel 2808  df-ne 2939  df-nel 3045  df-rex 3069  df-pss 3966
This theorem is referenced by:  sgrpssmgm  18850  mndsssgrp  18851
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