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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 41832 | . 2 ⊢ ({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵)) | |
2 | prid2g 4723 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐴, 𝐵}) | |
3 | 2 | 3ad2ant2 1135 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ {𝐴, 𝐵}) |
4 | necom 2998 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
5 | nelsn 4627 | . . . . 5 ⊢ (𝐵 ≠ 𝐴 → ¬ 𝐵 ∈ {𝐴}) | |
6 | 4, 5 | sylbi 216 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐵 ∈ {𝐴}) |
7 | 6 | 3ad2ant3 1136 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵) → ¬ 𝐵 ∈ {𝐴}) |
8 | 3, 7 | eldifd 3922 | . 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 1088 ∈ wcel 2107 ≠ wne 2944 Vcvv 3446 ∖ cdif 3908 {csn 4587 {cpr 4589 class class class wbr 5106 2oc2o 8407 ≈ cen 8881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1o 8413 df-2o 8414 df-en 8885 |
This theorem is referenced by: (None) |
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