![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5423 | . . 3 ⊢ {𝐴, 𝐵} ∈ V | |
2 | 1 | prid2 4760 | . 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}} |
3 | opi1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | opi1.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | dfop 4865 | . 2 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} |
6 | 2, 5 | eleqtrri 2824 | 1 ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3466 {csn 4621 {cpr 4623 ⟨cop 4627 |
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-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 |
This theorem is referenced by: opeluu 5461 uniopel 5507 elvvuni 5743 |
Copyright terms: Public domain | W3C validator |