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Mathbox for Richard Penner |
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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 43048 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵)) | |
2 | prid1g 4760 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵}) | |
3 | 2 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ {𝐴, 𝐵}) |
4 | nelsn 4664 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵}) | |
5 | 4 | 3ad2ant3 1132 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 ∈ {𝐵}) |
6 | 3, 5 | eldifd 3950 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵})) |
7 | 1, 6 | sylbi 216 | 1 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2930 Vcvv 3463 ∖ cdif 3936 {csn 4624 {cpr 4626 class class class wbr 5143 2oc2o 8479 ≈ cen 8959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1o 8485 df-2o 8486 df-en 8963 |
This theorem is referenced by: (None) |
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