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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 5390 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
2 | 1 | prid2 4725 | . 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}} |
3 | opi1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | opi1.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | dfop 4830 | . 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} |
6 | 2, 5 | eleqtrri 2833 | 1 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3444 {csn 4587 {cpr 4589 ⟨cop 4593 |
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-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 |
This theorem is referenced by: opeluu 5428 uniopel 5474 elvvuni 5709 |
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