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Mirrors > Home > MPE Home > Th. List > eldifpr | Structured version Visualization version GIF version |
Description: Membership in a set with two elements removed. Similar to eldifsn 4752 and eldiftp 4652. (Contributed by Mario Carneiro, 18-Jul-2017.) |
Ref | Expression |
---|---|
eldifpr | ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprg 4612 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) | |
2 | 1 | notbid 318 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ ¬ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) |
3 | neanior 3038 | . . . 4 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) | |
4 | 2, 3 | bitr4di 289 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))) |
5 | 4 | pm5.32i 576 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))) |
6 | eldif 3925 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷})) | |
7 | 3anass 1096 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))) | |
8 | 5, 6, 7 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∖ cdif 3912 {cpr 4593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-v 3450 df-dif 3918 df-un 3920 df-sn 4592 df-pr 4594 |
This theorem is referenced by: rexdifpr 4624 logbcl 26133 logbid1 26134 logb1 26135 elogb 26136 logbchbase 26137 relogbval 26138 relogbcl 26139 relogbreexp 26141 relogbmul 26143 relogbexp 26146 nnlogbexp 26147 relogbcxp 26151 cxplogb 26152 relogbcxpb 26153 logbmpt 26154 logbfval 26156 logbgt0b 26159 2logb9irrALT 26164 sqrt2cxp2logb9e3 26165 neldifpr1 31502 neldifpr2 31503 eluz2cnn0n1 46666 rege1logbrege0 46718 relogbmulbexp 46721 relogbdivb 46722 nnpw2blen 46740 |
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