![]() |
Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
|
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 2955 | . . . 4 ⊢ Ⅎ𝑧V | |
2 | nfab1 2957 | . . . 4 ⊢ Ⅎ𝑧{𝑧 ∣ 𝜑} | |
3 | 1, 2 | nfdif 4053 | . . 3 ⊢ Ⅎ𝑧(V ∖ {𝑧 ∣ 𝜑}) |
4 | nfab1 2957 | . . 3 ⊢ Ⅎ𝑧{𝑧 ∣ ¬ 𝜑} | |
5 | 3, 4 | cleqf 2983 | . 2 ⊢ ((V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} ↔ ∀𝑧(𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑})) |
6 | abid 2780 | . . . 4 ⊢ (𝑧 ∈ {𝑧 ∣ 𝜑} ↔ 𝜑) | |
7 | 6 | notbii 323 | . . 3 ⊢ (¬ 𝑧 ∈ {𝑧 ∣ 𝜑} ↔ ¬ 𝜑) |
8 | velcomp 3896 | . . 3 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ ¬ 𝑧 ∈ {𝑧 ∣ 𝜑}) | |
9 | abid 2780 | . . 3 ⊢ (𝑧 ∈ {𝑧 ∣ ¬ 𝜑} ↔ ¬ 𝜑) | |
10 | 7, 8, 9 | 3bitr4i 306 | . 2 ⊢ (𝑧 ∈ (V ∖ {𝑧 ∣ 𝜑}) ↔ 𝑧 ∈ {𝑧 ∣ ¬ 𝜑}) |
11 | 5, 10 | mpgbir 1801 | 1 ⊢ (V ∖ {𝑧 ∣ 𝜑}) = {𝑧 ∣ ¬ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2776 Vcvv 3441 ∖ cdif 3878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |