|   | 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 4309 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)} | |
| 2 | difin 4271 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) = (𝐴 ∖ {𝑥 ∣ 𝜑}) | |
| 3 | dfrab3 4318 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
| 4 | 3 | difeq2i 4122 | . . 3 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) | 
| 5 | abid2 2878 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
| 6 | 5 | difeq1i 4121 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = (𝐴 ∖ {𝑥 ∣ 𝜑}) | 
| 7 | 2, 4, 6 | 3eqtr4i 2774 | . 2 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) | 
| 8 | df-rab 3436 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)} | |
| 9 | 1, 7, 8 | 3eqtr4i 2774 | 1 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 {crab 3435 ∖ cdif 3947 ∩ cin 3949 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-in 3957 | 
| This theorem is referenced by: rlimrege0 15616 mhpmulcl 22154 ordtcld1 23206 ordtcld2 23207 lhop1lem 26053 rpvmasumlem 27532 finsumvtxdg2ssteplem1 29564 frgrwopreglem3 30334 zarcls 33874 hasheuni 34087 braew 34244 satfv1 35369 | 
| Copyright terms: Public domain | W3C validator |