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Theorem ssexnelpss 4028
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 3047 . . . 4 (𝑥𝐴 ↔ ¬ 𝑥𝐴)
2 ssnelpss 4026 . . . . 5 (𝐴𝐵 → ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → 𝐴𝐵))
32expdimp 456 . . . 4 ((𝐴𝐵𝑥𝐵) → (¬ 𝑥𝐴𝐴𝐵))
41, 3syl5bi 245 . . 3 ((𝐴𝐵𝑥𝐵) → (𝑥𝐴𝐴𝐵))
54rexlimdva 3203 . 2 (𝐴𝐵 → (∃𝑥𝐵 𝑥𝐴𝐴𝐵))
65imp 410 1 ((𝐴𝐵 ∧ ∃𝑥𝐵 𝑥𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wcel 2110  wnel 3046  wrex 3062  wss 3866  wpss 3867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-pss 3885
This theorem is referenced by:  sgrpssmgm  18360  mndsssgrp  18361
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