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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 4114 | . . . . 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 3155 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴 → 𝐴 ⊊ 𝐵)) |
| 6 | 5 | imp 406 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 ∉ wnel 3046 ∃wrex 3070 ⊆ wss 3951 ⊊ wpss 3952 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-rex 3071 df-pss 3971 |
| This theorem is referenced by: sgrpssmgm 18946 mndsssgrp 18947 |
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