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Mathbox for Andrew Salmon |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > compab | Structured version Visualization version GIF version |
Description: Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
compab | ⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑧V | |
2 | nfab1 2906 | . . . 4 ⊢ Ⅎ𝑧{𝑧 ∣ 𝜑} | |
3 | 1, 2 | nfdif 4146 | . . 3 ⊢ Ⅎ𝑧(V ∖ {𝑧 ∣ 𝜑}) |
4 | nfab1 2906 | . . 3 ⊢ Ⅎ𝑧{𝑧 ∣ ¬ 𝜑} | |
5 | 3, 4 | cleqf 2936 | . 2 ⊢ ((V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})) |
6 | abid 2715 | . . . 4 ⊢ (𝑧 ∈ {𝑧 ∣ 𝜑} ↔ 𝜑) | |
7 | 6 | notbii 320 | . . 3 ⊢ (¬ 𝑧 ∈ {𝑧 ∣ 𝜑} ↔ ¬ 𝜑) |
8 | velcomp 3985 | . . 3 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ ¬ 𝑧 ∈ {𝑧 ∣ 𝜑}) | |
9 | abid 2715 | . . 3 ⊢ (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑) | |
10 | 7, 8, 9 | 3bitr4i 303 | . 2 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}) |
11 | 5, 10 | mpgbir 1797 | 1 ⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1537 ∈ wcel 2103 {cab 2711 Vcvv 3482 ∖ cdif 3967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-v 3484 df-dif 3973 |
This theorem is referenced by: (None) |
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