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Theorem ssexnelpss 4125
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 3044 . . . 4 (𝑥𝐴 ↔ ¬ 𝑥𝐴)
2 ssnelpss 4123 . . . . 5 (𝐴𝐵 → ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → 𝐴𝐵))
32expdimp 452 . . . 4 ((𝐴𝐵𝑥𝐵) → (¬ 𝑥𝐴𝐴𝐵))
41, 3biimtrid 242 . . 3 ((𝐴𝐵𝑥𝐵) → (𝑥𝐴𝐴𝐵))
54rexlimdva 3152 . 2 (𝐴𝐵 → (∃𝑥𝐵 𝑥𝐴𝐴𝐵))
65imp 406 1 ((𝐴𝐵 ∧ ∃𝑥𝐵 𝑥𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  wnel 3043  wrex 3067  wss 3962  wpss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726  df-clel 2813  df-ne 2938  df-nel 3044  df-rex 3068  df-pss 3982
This theorem is referenced by:  sgrpssmgm  18958  mndsssgrp  18959
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