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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 3912 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
2 | 1 | biantrur 528 | . . 3 ⊢ (¬ (𝐴 ∪ 𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) |
3 | ssid 3757 | . . . . 5 ⊢ 𝐵 ⊆ 𝐵 | |
4 | 3 | biantru 527 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) |
5 | unss 3922 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵) | |
6 | 4, 5 | bitri 264 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵) |
7 | 2, 6 | xchnxbir 322 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) |
8 | dfpss3 3827 | . 2 ⊢ (𝐵 ⊊ (𝐴 ∪ 𝐵) ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) | |
9 | 7, 8 | bitr4i 267 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 383 ∪ cun 3705 ⊆ wss 3707 ⊊ wpss 3708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-v 3334 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 |
This theorem is referenced by: disjpss 4164 lindsenlbs 33709 |
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