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Theorem ssexnelpss 3827
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 3000 . . . 4 (𝑥𝐴 ↔ ¬ 𝑥𝐴)
2 ssnelpss 3825 . . . . 5 (𝐴𝐵 → ((𝑥𝐵 ∧ ¬ 𝑥𝐴) → 𝐴𝐵))
32expdimp 452 . . . 4 ((𝐴𝐵𝑥𝐵) → (¬ 𝑥𝐴𝐴𝐵))
41, 3syl5bi 232 . . 3 ((𝐴𝐵𝑥𝐵) → (𝑥𝐴𝐴𝐵))
54rexlimdva 3133 . 2 (𝐴𝐵 → (∃𝑥𝐵 𝑥𝐴𝐴𝐵))
65imp 444 1 ((𝐴𝐵 ∧ ∃𝑥𝐵 𝑥𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2103  wnel 2999  wrex 3015  wss 3680  wpss 3681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1818  df-cleq 2717  df-clel 2720  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-pss 3696
This theorem is referenced by:  sgrpssmgm  17542  mndsssgrp  17543
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