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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 3975 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
2 | 1 | biantrur 527 | . . 3 ⊢ (¬ (𝐴 ∪ 𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) |
3 | ssid 3819 | . . . . 5 ⊢ 𝐵 ⊆ 𝐵 | |
4 | 3 | biantru 526 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) |
5 | unss 3985 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵) | |
6 | 4, 5 | bitri 267 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵) |
7 | 2, 6 | xchnxbir 325 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) |
8 | dfpss3 3890 | . 2 ⊢ (𝐵 ⊊ (𝐴 ∪ 𝐵) ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) | |
9 | 7, 8 | bitr4i 270 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 385 ∪ cun 3767 ⊆ wss 3769 ⊊ wpss 3770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-v 3387 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 |
This theorem is referenced by: disjpss 4223 lindsenlbs 33893 |
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