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Mirrors > Home > MPE Home > Th. List > enpr2 | Structured version Visualization version GIF version |
Description: An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 9088. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5371, ax-un 7754. (Revised by BTernaryTau, 30-Dec-2024.) |
Ref | Expression |
---|---|
enpr2 | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2939 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | simp1 1135 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶) | |
3 | simp2 1136 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝐷) | |
4 | simp3 1137 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
5 | 2, 3, 4 | enpr2d 9088 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ≈ 2o) |
6 | 1, 5 | syl3an3b 1404 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {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-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-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-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-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-1o 8505 df-2o 8506 df-en 8985 |
This theorem is referenced by: pr2ne 10042 pr2neOLD 10043 en2eqpr 10045 en2eleq 10046 pr2pwpr 14515 pmtrprfv 19486 pmtrprfv3 19487 symggen 19503 pmtr3ncomlem1 19506 pmtr3ncom 19508 mdetralt 22630 en2top 23008 hmphindis 23821 pmtrcnel 33092 pmtrcnel2 33093 fzo0pmtrlast 33095 pmtridf1o 33097 pmtrto1cl 33102 cycpm2tr 33122 cyc3evpm 33153 cyc3genpmlem 33154 cyc3conja 33160 |
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