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Mirrors > Home > MPE Home > Th. List > opi2 | Structured version Visualization version GIF version |
Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
opi1.1 | ⊢ 𝐴 ∈ V |
opi1.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opi2 | ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5325 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
2 | 1 | prid2 4679 | . 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}} |
3 | opi1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | opi1.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | dfop 4783 | . 2 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
6 | 2, 5 | eleqtrri 2837 | 1 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3408 {csn 4541 {cpr 4543 〈cop 4547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 |
This theorem is referenced by: opeluu 5354 uniopel 5399 elvvuni 5625 |
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