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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pr2eldif1 | Structured version Visualization version GIF version | ||
| Description: If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| pr2eldif1 | ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pren2 43571 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵)) | |
| 2 | prid1g 4759 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵}) | |
| 3 | 2 | 3ad2ant1 1133 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ {𝐴, 𝐵}) | 
| 4 | nelsn 4665 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵}) | |
| 5 | 4 | 3ad2ant3 1135 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 ∈ {𝐵}) | 
| 6 | 3, 5 | eldifd 3961 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵})) | 
| 7 | 1, 6 | sylbi 217 | 1 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∖ cdif 3947 {csn 4625 {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-11 2156 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-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 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 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-1o 8507 df-2o 8508 df-en 8987 | 
| This theorem is referenced by: (None) | 
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