Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dmsnm | GIF version |
Description: The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
Ref | Expression |
---|---|
dmsnm | ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elvv 4601 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eldm 4736 | . . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦) |
4 | df-br 3930 | . . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {𝐴}) | |
5 | vex 2689 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
6 | 2, 5 | opex 4151 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V |
7 | 6 | elsn 3543 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {𝐴} ↔ 〈𝑥, 𝑦〉 = 𝐴) |
8 | eqcom 2141 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 ↔ 𝐴 = 〈𝑥, 𝑦〉) | |
9 | 4, 7, 8 | 3bitri 205 | . . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = 〈𝑥, 𝑦〉) |
10 | 9 | exbii 1584 | . . . 4 ⊢ (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
11 | 3, 10 | bitr2i 184 | . . 3 ⊢ (∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ 𝑥 ∈ dom {𝐴}) |
12 | 11 | exbii 1584 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
13 | 1, 12 | bitri 183 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 {csn 3527 〈cop 3530 class class class wbr 3929 × cxp 4537 dom cdm 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-dm 4549 |
This theorem is referenced by: rnsnm 5005 dmsn0 5006 dmsn0el 5008 relsn2m 5009 |
Copyright terms: Public domain | W3C validator |