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Theorem ssexnelpss 3704
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 2894 . . . 4 (𝑥𝐴 ↔ ¬ 𝑥𝐴)
2 ssnelpss 3702 . . . . 5 (𝐴𝐵 → ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → 𝐴𝐵))
32expdimp 453 . . . 4 ((𝐴𝐵𝑥𝐵) → (¬ 𝑥𝐴𝐴𝐵))
41, 3syl5bi 232 . . 3 ((𝐴𝐵𝑥𝐵) → (𝑥𝐴𝐴𝐵))
54rexlimdva 3026 . 2 (𝐴𝐵 → (∃𝑥𝐵 𝑥𝐴𝐴𝐵))
65imp 445 1 ((𝐴𝐵 ∧ ∃𝑥𝐵 𝑥𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 1987  wnel 2893  wrex 2909  wss 3560  wpss 3561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-cleq 2614  df-clel 2617  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-pss 3576
This theorem is referenced by:  sgrpssmgm  17360  mndsssgrp  17361
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