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Mirrors > Home > MPE Home > Th. List > notab | Structured version Visualization version GIF version |
Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.) |
Ref | Expression |
---|---|
notab | ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2921 | . . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} | |
2 | rabab 3223 | . . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑} | |
3 | 1, 2 | eqtr3i 2646 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑} |
4 | difab 3896 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} | |
5 | abid2 2745 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ V} = V | |
6 | 5 | difeq1i 3724 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = (V ∖ {𝑥 ∣ 𝜑}) |
7 | 4, 6 | eqtr3i 2646 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥 ∣ 𝜑}) |
8 | 3, 7 | eqtr3i 2646 | 1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 {crab 2916 Vcvv 3200 ∖ cdif 3571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 |
This theorem is referenced by: dfif3 4100 |
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