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Mirrors > Home > MPE Home > Th. List > dmsnn0 | Structured version Visualization version GIF version |
Description: The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmsnn0 | ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3478 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm 5900 | . . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦) |
3 | df-br 5149 | . . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴}) | |
4 | opex 5464 | . . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V | |
5 | 4 | elsn 4643 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴) |
6 | eqcom 2739 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 ↔ 𝐴 = ⟨𝑥, 𝑦⟩) | |
7 | 3, 5, 6 | 3bitri 296 | . . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = ⟨𝑥, 𝑦⟩) |
8 | 7 | exbii 1850 | . . . 4 ⊢ (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) |
9 | 2, 8 | bitr2i 275 | . . 3 ⊢ (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴}) |
10 | 9 | exbii 1850 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
11 | elvv 5750 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) | |
12 | n0 4346 | . 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
13 | 10, 11, 12 | 3bitr4i 302 | 1 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∅c0 4322 {csn 4628 ⟨cop 4634 class class class wbr 5148 × cxp 5674 dom cdm 5676 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-dm 5686 |
This theorem is referenced by: rnsnn0 6207 dmsn0 6208 dmsn0el 6210 relsn2 6211 1stnpr 7978 1st2val 8002 mpoxopxnop0 8199 cnvfi 9179 hashfun 14396 fineqvac 34092 |
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