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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 5334 | . . 3 ⊢ {𝐴} ∈ V | |
2 | 1 | prid1 4700 | . 2 ⊢ {𝐴} ∈ {{𝐴}, {𝐴, 𝐵}} |
3 | opi1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | opi1.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | dfop 4804 | . 2 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
6 | 2, 5 | eleqtrri 2914 | 1 ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 Vcvv 3496 {csn 4569 {cpr 4571 〈cop 4575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 |
This theorem is referenced by: opth1 5369 opth 5370 |
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