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| Description: An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 9090. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5364, ax-un 7756. (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 2940 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
| 2 | simp1 1136 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶) | |
| 3 | simp2 1137 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝐷) | |
| 4 | simp3 1138 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
| 5 | 2, 3, 4 | enpr2d 9090 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → {𝐴, 𝐵} ≈ 2o) | 
| 6 | 1, 5 | syl3an3b 1406 | 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 1539 ∈ wcel 2107 ≠ wne 2939 {cpr 4627 class class class wbr 5142 2oc2o 8501 ≈ cen 8983 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-suc 6389 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-1o 8507 df-2o 8508 df-en 8987 | 
| This theorem is referenced by: pr2ne 10045 pr2neOLD 10046 en2eqpr 10048 en2eleq 10049 pr2pwpr 14519 pmtrprfv 19472 pmtrprfv3 19473 symggen 19489 pmtr3ncomlem1 19492 pmtr3ncom 19494 mdetralt 22615 en2top 22993 hmphindis 23806 pmtrcnel 33110 pmtrcnel2 33111 fzo0pmtrlast 33113 pmtridf1o 33115 pmtrto1cl 33120 cycpm2tr 33140 cyc3evpm 33171 cyc3genpmlem 33172 cyc3conja 33178 | 
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