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Mirrors > Home > MPE Home > Th. List > opi1 | Structured version Visualization version GIF version |
Description: One of the two elements in an ordered pair. (Contributed by NM, 15-Jul-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 |
---|---|
opi1 | ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5331 | . . 3 ⊢ {𝐴} ∈ V | |
2 | 1 | prid1 4697 | . 2 ⊢ {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}} |
3 | opi1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | opi1.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | dfop 4801 | . 2 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
6 | 2, 5 | eleqtrri 2912 | 1 ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3494 {csn 4566 {cpr 4568 〈cop 4572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 |
This theorem is referenced by: opth1 5366 opth 5367 |
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