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Mirrors > Home > MPE Home > Th. List > ssexnelpss | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ssexnelpss | ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3047 | . . . 4 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴) | |
2 | ssnelpss 4072 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) | |
3 | 2 | expdimp 454 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (¬ 𝑥 ∈ 𝐴 → 𝐴 ⊊ 𝐵)) |
4 | 1, 3 | biimtrid 241 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∉ 𝐴 → 𝐴 ⊊ 𝐵)) |
5 | 4 | rexlimdva 3149 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴 → 𝐴 ⊊ 𝐵)) |
6 | 5 | imp 408 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∈ wcel 2107 ∉ wnel 3046 ∃wrex 3070 ⊆ wss 3911 ⊊ wpss 3912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-clel 2811 df-ne 2941 df-nel 3047 df-rex 3071 df-pss 3930 |
This theorem is referenced by: sgrpssmgm 18748 mndsssgrp 18749 |
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