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Mirrors > Home > MPE Home > Th. List > disjpss | Structured version Visualization version GIF version |
Description: A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
disjpss | ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3999 | . . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵 | |
2 | 1 | biantru 529 | . . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵)) |
3 | ssin 4225 | . . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵)) | |
4 | 2, 3 | bitri 275 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵)) |
5 | sseq2 4003 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∩ 𝐵) ↔ 𝐵 ⊆ ∅)) | |
6 | 4, 5 | bitrid 283 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ ∅)) |
7 | ss0 4393 | . . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅) | |
8 | 6, 7 | biimtrdi 252 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 → 𝐵 = ∅)) |
9 | 8 | necon3ad 2947 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ 𝐴)) |
10 | 9 | imp 406 | . 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ 𝐴) |
11 | nsspssun 4252 | . . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐵 ∪ 𝐴)) | |
12 | uncom 4148 | . . . 4 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
13 | 12 | psseq2i 4085 | . . 3 ⊢ (𝐴 ⊊ (𝐵 ∪ 𝐴) ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵)) |
14 | 11, 13 | bitri 275 | . 2 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵)) |
15 | 10, 14 | sylib 217 | 1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ≠ wne 2934 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 ⊊ wpss 3944 ∅c0 4317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 |
This theorem is referenced by: omsucne 7870 isfin1-3 10380 |
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