![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elid3 | Structured version Visualization version GIF version |
Description: Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.) |
Ref | Expression |
---|---|
bj-elid3 | ⊢ (〈𝑥, 𝐴〉 ∈ I ↔ 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3444 | . 2 ⊢ 𝑥 ∈ V | |
2 | bj-opelidb1 34568 | . 2 ⊢ (〈𝑥, 𝐴〉 ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴)) | |
3 | 1, 2 | mpbiran 708 | 1 ⊢ (〈𝑥, 𝐴〉 ∈ I ↔ 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 I cid 5424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-id 5425 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |