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Mirrors > Home > HSE Home > Th. List > chocnul | Structured version Visualization version GIF version |
Description: Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chocnul | ⊢ (⊥‘∅) = ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4109 | . . 3 ⊢ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0 | |
2 | 0ss 4005 | . . . 4 ⊢ ∅ ⊆ ℋ | |
3 | ocel 28268 | . . . 4 ⊢ (∅ ⊆ ℋ → (𝑥 ∈ (⊥‘∅) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0))) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝑥 ∈ (⊥‘∅) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0)) |
5 | 1, 4 | mpbiran2 974 | . 2 ⊢ (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ) |
6 | 5 | eqriv 2648 | 1 ⊢ (⊥‘∅) = ℋ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ⊆ wss 3607 ∅c0 3948 ‘cfv 5926 (class class class)co 6690 0cc0 9974 ℋchil 27904 ·ih csp 27907 ⊥cort 27915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-hilex 27984 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-oc 28237 |
This theorem is referenced by: (None) |
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