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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 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | eldm 5769 | . . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦) |
3 | df-br 5067 | . . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {𝐴}) | |
4 | opex 5356 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
5 | 4 | elsn 4582 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {𝐴} ↔ 〈𝑥, 𝑦〉 = 𝐴) |
6 | eqcom 2828 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 ↔ 𝐴 = 〈𝑥, 𝑦〉) | |
7 | 3, 5, 6 | 3bitri 299 | . . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = 〈𝑥, 𝑦〉) |
8 | 7 | exbii 1848 | . . . 4 ⊢ (∃𝑦 𝑥{𝐴}𝑦 ↔ ∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
9 | 2, 8 | bitr2i 278 | . . 3 ⊢ (∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ 𝑥 ∈ dom {𝐴}) |
10 | 9 | exbii 1848 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
11 | elvv 5626 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
12 | n0 4310 | . 2 ⊢ (dom {𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
13 | 10, 11, 12 | 3bitr4i 305 | 1 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 {csn 4567 〈cop 4573 class class class wbr 5066 × cxp 5553 dom cdm 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-dm 5565 |
This theorem is referenced by: rnsnn0 6065 dmsn0 6066 dmsn0el 6068 relsn2 6069 1stnpr 7693 1st2val 7717 mpoxopxnop0 7881 hashfun 13799 |
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