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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 43032 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵)) | |
2 | prid2g 4770 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐴, 𝐵}) | |
3 | 2 | 3ad2ant2 1131 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ {𝐴, 𝐵}) |
4 | necom 2991 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
5 | nelsn 4673 | . . . . 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 3960 | . 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 2937 Vcvv 3473 ∖ cdif 3946 {csn 4632 {cpr 4634 class class class wbr 5152 2oc2o 8489 ≈ cen 8969 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-1o 8495 df-2o 8496 df-en 8973 |
This theorem is referenced by: (None) |
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