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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 4455 | . . 3 ⊢ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0 | |
2 | 0ss 4349 | . . . 4 ⊢ ∅ ⊆ ℋ | |
3 | ocel 29052 | . . . 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 708 | . 2 ⊢ (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ) |
6 | 5 | eqriv 2818 | 1 ⊢ (⊥‘∅) = ℋ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 ‘cfv 6349 (class class class)co 7150 0cc0 10531 ℋchba 28690 ·ih csp 28693 ⊥cort 28701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-hilex 28770 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oc 29023 |
This theorem is referenced by: (None) |
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