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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 4151 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
2 | 1 | biantrur 533 | . . 3 ⊢ (¬ (𝐴 ∪ 𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) |
3 | ssid 3991 | . . . . 5 ⊢ 𝐵 ⊆ 𝐵 | |
4 | 3 | biantru 532 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) |
5 | unss 4162 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵) | |
6 | 4, 5 | bitri 277 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) ⊆ 𝐵) |
7 | 2, 6 | xchnxbir 335 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) |
8 | dfpss3 4065 | . 2 ⊢ (𝐵 ⊊ (𝐴 ∪ 𝐵) ↔ (𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ ¬ (𝐴 ∪ 𝐵) ⊆ 𝐵)) | |
9 | 7, 8 | bitr4i 280 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ 𝐵 ⊊ (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∪ cun 3936 ⊆ wss 3938 ⊊ wpss 3939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 |
This theorem is referenced by: disjpss 4412 lindsenlbs 34889 |
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