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Mirrors > Home > MPE Home > Th. List > Mathboxes > pr2el1 | Structured version Visualization version GIF version |
Description: If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.) |
Ref | Expression |
---|---|
pr2el1 | ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pr2cv 43539 | . . 3 ⊢ ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | simpld 494 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ V) |
3 | prid1g 4758 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵}) | |
4 | 2, 3 | syl 17 | 1 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3479 {cpr 4626 class class class wbr 5141 2oc2o 8496 ≈ cen 8978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ord 6385 df-on 6386 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-1o 8502 df-2o 8503 df-en 8982 |
This theorem is referenced by: (None) |
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