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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 4609 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | eldm 4744 | . . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦) |
4 | df-br 3938 | . . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {𝐴}) | |
5 | vex 2692 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
6 | 2, 5 | opex 4159 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V |
7 | 6 | elsn 3548 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {𝐴} ↔ 〈𝑥, 𝑦〉 = 𝐴) |
8 | eqcom 2142 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 ↔ 𝐴 = 〈𝑥, 𝑦〉) | |
9 | 4, 7, 8 | 3bitri 205 | . . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = 〈𝑥, 𝑦〉) |
10 | 9 | exbii 1585 | . . . 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 1585 | . 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 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 {csn 3532 〈cop 3535 class class class wbr 3937 × cxp 4545 dom cdm 4547 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-dm 4557 |
This theorem is referenced by: rnsnm 5013 dmsn0 5014 dmsn0el 5016 relsn2m 5017 |
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