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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 3773 | . . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵 | |
2 | 1 | biantru 519 | . . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵)) |
3 | ssin 3983 | . . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵)) | |
4 | 2, 3 | bitri 264 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵)) |
5 | sseq2 3776 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ (𝐴 ∩ 𝐵) ↔ 𝐵 ⊆ ∅)) | |
6 | 4, 5 | syl5bb 272 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ ∅)) |
7 | ss0 4119 | . . . . 5 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅) | |
8 | 6, 7 | syl6bi 243 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ⊆ 𝐴 → 𝐵 = ∅)) |
9 | 8 | necon3ad 2956 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ 𝐴)) |
10 | 9 | imp 393 | . 2 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵 ⊆ 𝐴) |
11 | nsspssun 4006 | . . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐵 ∪ 𝐴)) | |
12 | uncom 3908 | . . . 4 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
13 | 12 | psseq2i 3847 | . . 3 ⊢ (𝐴 ⊊ (𝐵 ∪ 𝐴) ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵)) |
14 | 11, 13 | bitri 264 | . 2 ⊢ (¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ (𝐴 ∪ 𝐵)) |
15 | 10, 14 | sylib 208 | 1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1631 ≠ wne 2943 ∪ cun 3721 ∩ cin 3722 ⊆ wss 3723 ⊊ wpss 3724 ∅c0 4063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 |
This theorem is referenced by: isfin1-3 9414 |
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