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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 3045 | . . . 4 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴) | |
2 | ssnelpss 4110 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴) → 𝐴 ⊊ 𝐵)) | |
3 | 2 | expdimp 451 | . . . 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 3153 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴 → 𝐴 ⊊ 𝐵)) |
6 | 5 | imp 405 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2104 ∉ wnel 3044 ∃wrex 3068 ⊆ wss 3947 ⊊ wpss 3948 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-cleq 2722 df-clel 2808 df-ne 2939 df-nel 3045 df-rex 3069 df-pss 3966 |
This theorem is referenced by: sgrpssmgm 18850 mndsssgrp 18851 |
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