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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 4513 | . . 3 ⊢ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0 | |
| 2 | 0ss 4400 | . . . 4 ⊢ ∅ ⊆ ℋ | |
| 3 | ocel 31300 | . . . 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 710 | . 2 ⊢ (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ) |
| 6 | 5 | eqriv 2734 | 1 ⊢ (⊥‘∅) = ℋ |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℋchba 30938 ·ih csp 30941 ⊥cort 30949 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hilex 31018 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oc 31271 |
| This theorem is referenced by: (None) |
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