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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 4786 and eldiftp 4686. (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 4645 | . . . . 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 3031 | . . . 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 574 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))) |
6 | eldif 3955 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶, 𝐷})) | |
7 | 3anass 1093 | . 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 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∖ cdif 3942 {cpr 4626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-v 3472 df-dif 3948 df-un 3950 df-sn 4625 df-pr 4627 |
This theorem is referenced by: rexdifpr 4657 logbcl 26692 logbid1 26693 logb1 26694 elogb 26695 logbchbase 26696 relogbval 26697 relogbcl 26698 relogbreexp 26700 relogbmul 26702 relogbexp 26705 nnlogbexp 26706 relogbcxp 26710 cxplogb 26711 relogbcxpb 26712 logbmpt 26713 logbfval 26715 logbgt0b 26718 2logb9irrALT 26723 sqrt2cxp2logb9e3 26724 neldifpr1 32322 neldifpr2 32323 eluz2cnn0n1 47573 rege1logbrege0 47625 relogbmulbexp 47628 relogbdivb 47629 nnpw2blen 47647 |
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