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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 5362, ax-un 7721. (Revised by BTernaryTau, 30-Dec-2024.) |
Ref | Expression |
---|---|
pr2ne | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snnen2o 9233 | . . . 4 ⊢ ¬ {𝐴} ≈ 2o | |
2 | dfsn2 4640 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | preq2 4737 | . . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
4 | 2, 3 | eqtr2id 2785 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | breq1d 5157 | . . . 4 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} ≈ 2o ↔ {𝐴} ≈ 2o)) |
6 | 1, 5 | mtbiri 326 | . . 3 ⊢ (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2o) |
7 | 6 | necon2ai 2970 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ≠ 𝐵) |
8 | enpr2 9993 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) | |
9 | 8 | 3expia 1121 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2o)) |
10 | 7, 9 | impbid2 225 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 {csn 4627 {cpr 4629 class class class wbr 5147 2oc2o 8456 ≈ cen 8932 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1o 8462 df-2o 8463 df-en 8936 |
This theorem is referenced by: prdom2 9997 isprm2lem 16614 pmtrrn2 19322 mdetunilem7 22111 trsp2cyc 32269 en2pr 42283 pr2cv 42284 pren2 42289 |
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