Theorem ssexnelpss 3720

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 2898	. . . 4 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
2		ssnelpss 3718	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴) → 𝐴 ⊊ 𝐵))
3	2	expdimp 453	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (¬ 𝑥 ∈ 𝐴 → 𝐴 ⊊ 𝐵))
4	1, 3	syl5bi 232	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∉ 𝐴 → 𝐴 ⊊ 𝐵))
5	4	rexlimdva 3031	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴 → 𝐴 ⊊ 𝐵))
6	5	imp 445	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∈ wcel 1990   ∉ wnel 2897  ∃wrex 2913   ⊆ wss 3574   ⊊ wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-pss 3590

This theorem is referenced by:  sgrpssmgm  17420  mndsssgrp  17421