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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 5428 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
2 | 1 | prid2 4763 | . 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}} |
3 | opi1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | opi1.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | dfop 4868 | . 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} |
6 | 2, 5 | eleqtrri 2828 | 1 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 Vcvv 3470 {csn 4624 {cpr 4626 ⟨cop 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 |
This theorem is referenced by: opeluu 5466 uniopel 5512 elvvuni 5748 |
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