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Mirrors > Home > MPE Home > Th. List > nssss | Structured version Visualization version GIF version |
Description: Negation of subclass relationship. Compare nss 3912. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
nssss | ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exanali 1822 | . . 3 ⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ ¬ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) | |
2 | ssextss 5199 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) | |
3 | 1, 2 | xchbinxr 327 | . 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) ↔ ¬ 𝐴 ⊆ 𝐵) |
4 | 3 | bicomi 216 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∀wal 1506 ∃wex 1743 ⊆ wss 3822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-pw 4418 df-sn 4436 df-pr 4438 |
This theorem is referenced by: (None) |
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