![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > notrab | Structured version Visualization version GIF version |
Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
notrab | ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difab 4224 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)} | |
2 | difin 4188 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) = (𝐴 ∖ {𝑥 ∣ 𝜑}) | |
3 | dfrab3 4230 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
4 | 3 | difeq2i 4047 | . . 3 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) |
5 | abid2 2932 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
6 | 5 | difeq1i 4046 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = (𝐴 ∖ {𝑥 ∣ 𝜑}) |
7 | 2, 4, 6 | 3eqtr4i 2831 | . 2 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) |
8 | df-rab 3115 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)} | |
9 | 1, 7, 8 | 3eqtr4i 2831 | 1 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 {crab 3110 ∖ cdif 3878 ∩ cin 3880 |
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-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-rab 3115 df-v 3443 df-dif 3884 df-in 3888 |
This theorem is referenced by: rlimrege0 14928 ordtcld1 21802 ordtcld2 21803 lhop1lem 24616 rpvmasumlem 26071 finsumvtxdg2ssteplem1 27335 frgrwopreglem3 28099 zarcls 31227 hasheuni 31454 braew 31611 satfv1 32723 |
Copyright terms: Public domain | W3C validator |