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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 9049. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5364, ax-un 7725. (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 2942 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | simp1 1137 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶) | |
3 | simp2 1138 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝐷) | |
4 | simp3 1139 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
5 | 2, 3, 4 | enpr2d 9049 | . 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 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {cpr 4631 class class class wbr 5149 2oc2o 8460 ≈ cen 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-suc 6371 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-1o 8466 df-2o 8467 df-en 8940 |
This theorem is referenced by: pr2ne 9999 pr2neOLD 10000 en2eqpr 10002 en2eleq 10003 pr2pwpr 14440 pmtrprfv 19321 pmtrprfv3 19322 symggen 19338 pmtr3ncomlem1 19341 pmtr3ncom 19343 mdetralt 22110 en2top 22488 hmphindis 23301 pmtrcnel 32250 pmtrcnel2 32251 pmtridf1o 32253 pmtrto1cl 32258 cycpm2tr 32278 cyc3evpm 32309 cyc3genpmlem 32310 cyc3conja 32316 |
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