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| Mirrors > Home > MPE Home > Th. List > pr2ne | Structured version Visualization version GIF version | ||
| Description: If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5365, ax-un 7755. (Revised by BTernaryTau, 30-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| pr2ne | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | snnen2o 9273 | . . . 4 ⊢ ¬ {𝐴} ≈ 2o | |
| 2 | dfsn2 4639 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
| 3 | preq2 4734 | . . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
| 4 | 2, 3 | eqtr2id 2790 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) | 
| 5 | 4 | breq1d 5153 | . . . 4 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} ≈ 2o ↔ {𝐴} ≈ 2o)) | 
| 6 | 1, 5 | mtbiri 327 | . . 3 ⊢ (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2o) | 
| 7 | 6 | necon2ai 2970 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ≠ 𝐵) | 
| 8 | enpr2 10042 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) | |
| 9 | 8 | 3expia 1122 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2o)) | 
| 10 | 7, 9 | impbid2 226 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {csn 4626 {cpr 4628 class class class wbr 5143 2oc2o 8500 ≈ cen 8982 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1o 8506 df-2o 8507 df-en 8986 | 
| This theorem is referenced by: prdom2 10046 isprm2lem 16718 pmtrrn2 19478 mdetunilem7 22624 trsp2cyc 33143 en2pr 43560 pr2cv 43561 pren2 43566 | 
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