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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 5371, ax-un 7754. (Revised by BTernaryTau, 30-Dec-2024.) |
Ref | Expression |
---|---|
pr2ne | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snnen2o 9271 | . . . 4 ⊢ ¬ {𝐴} ≈ 2o | |
2 | dfsn2 4644 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | preq2 4739 | . . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
4 | 2, 3 | eqtr2id 2788 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | 4 | breq1d 5158 | . . . 4 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐵} ≈ 2o ↔ {𝐴} ≈ 2o)) |
6 | 1, 5 | mtbiri 327 | . . 3 ⊢ (𝐴 = 𝐵 → ¬ {𝐴, 𝐵} ≈ 2o) |
7 | 6 | necon2ai 2968 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ≠ 𝐵) |
8 | enpr2 10040 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) | |
9 | 8 | 3expia 1120 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 {csn 4631 {cpr 4633 class class class wbr 5148 2oc2o 8499 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1o 8505 df-2o 8506 df-en 8985 |
This theorem is referenced by: prdom2 10044 isprm2lem 16715 pmtrrn2 19493 mdetunilem7 22640 trsp2cyc 33126 en2pr 43537 pr2cv 43538 pren2 43543 |
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