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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pr2eldif2 | 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 |
---|---|
pr2eldif2 | ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ ({𝐴, 𝐵} ∖ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pren2 42880 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵)) | |
2 | prid2g 4760 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐴, 𝐵}) | |
3 | 2 | 3ad2ant2 1131 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ {𝐴, 𝐵}) |
4 | necom 2988 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
5 | nelsn 4663 | . . . . 5 ⊢ (𝐵 ≠ 𝐴 → ¬ 𝐵 ∈ {𝐴}) | |
6 | 4, 5 | sylbi 216 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐵 ∈ {𝐴}) |
7 | 6 | 3ad2ant3 1132 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → ¬ 𝐵 ∈ {𝐴}) |
8 | 3, 7 | eldifd 3954 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ ({𝐴, 𝐵} ∖ {𝐴})) |
9 | 1, 8 | sylbi 216 | 1 ⊢ ({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ ({𝐴, 𝐵} ∖ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∖ cdif 3940 {csn 4623 {cpr 4625 class class class wbr 5141 2oc2o 8461 ≈ cen 8938 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-1o 8467 df-2o 8468 df-en 8942 |
This theorem is referenced by: (None) |
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