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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 3479 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm 5901 | . . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦) |
3 | df-br 5150 | . . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {𝐴}) | |
4 | opex 5465 | . . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V | |
5 | 4 | elsn 4644 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝐴} ↔ ⟨𝑥, 𝑦⟩ = 𝐴) |
6 | eqcom 2740 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 ↔ 𝐴 = ⟨𝑥, 𝑦⟩) | |
7 | 3, 5, 6 | 3bitri 297 | . . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = ⟨𝑥, 𝑦⟩) |
8 | 7 | exbii 1851 | . . . 4 ⊢ (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) |
9 | 2, 8 | bitr2i 276 | . . 3 ⊢ (∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝑥 ∈ dom {𝐴}) |
10 | 9 | exbii 1851 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
11 | elvv 5751 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) | |
12 | n0 4347 | . 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
13 | 10, 11, 12 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∅c0 4323 {csn 4629 ⟨cop 4635 class class class wbr 5149 × cxp 5675 dom cdm 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-dm 5687 |
This theorem is referenced by: rnsnn0 6208 dmsn0 6209 dmsn0el 6211 relsn2 6212 1stnpr 7979 1st2val 8003 mpoxopxnop0 8200 cnvfi 9180 hashfun 14397 fineqvac 34097 |
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