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Mirrors > Home > MPE Home > Th. List > nsspssun | Structured version Visualization version GIF version |
Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
nsspssun | ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 3739 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
2 | 1 | biantrur 526 | . . 3 ⊢ (¬ (𝐴 ∪ 𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) |
3 | ssid 3587 | . . . . 5 ⊢ 𝐵 ⊆ 𝐵 | |
4 | 3 | biantru 525 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) |
5 | unss 3749 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵) | |
6 | 4, 5 | bitri 263 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵) |
7 | 2, 6 | xchnxbir 322 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) |
8 | dfpss3 3655 | . 2 ⊢ (𝐵 ⊊ (𝐴 ∪ 𝐵) ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) | |
9 | 7, 8 | bitr4i 266 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 ∪ cun 3538 ⊆ wss 3540 ⊊ wpss 3541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 |
This theorem is referenced by: disjpss 3980 lindsenlbs 32574 |
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