Theorem ssexnelpss 4082

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	⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssexnelpss

Step	Hyp	Ref	Expression
1		df-nel 3031	. . . 4 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
2		ssnelpss 4080	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴) → 𝐴 ⊊ 𝐵))
3	2	expdimp 452	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (¬ 𝑥 ∈ 𝐴 → 𝐴 ⊊ 𝐵))
4	1, 3	biimtrid 242	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∉ 𝐴 → 𝐴 ⊊ 𝐵))
5	4	rexlimdva 3135	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴 → 𝐴 ⊊ 𝐵))
6	5	imp 406	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2109   ∉ wnel 3030  ∃wrex 3054   ⊆ wss 3917   ⊊ wpss 3918

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2722  df-clel 2804  df-ne 2927  df-nel 3031  df-rex 3055  df-pss 3937

This theorem is referenced by:  sgrpssmgm  18867  mndsssgrp  18868