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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 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | 1 | eldm 5911 | . . . 4 ⊢ (𝑥 ∈ dom {𝐴} ↔ ∃𝑦 𝑥{𝐴}𝑦) |
| 3 | df-br 5144 | . . . . . 6 ⊢ (𝑥{𝐴}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {𝐴}) | |
| 4 | opex 5469 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
| 5 | 4 | elsn 4641 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {𝐴} ↔ 〈𝑥, 𝑦〉 = 𝐴) |
| 6 | eqcom 2744 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 ↔ 𝐴 = 〈𝑥, 𝑦〉) | |
| 7 | 3, 5, 6 | 3bitri 297 | . . . . 5 ⊢ (𝑥{𝐴}𝑦 ↔ 𝐴 = 〈𝑥, 𝑦〉) |
| 8 | 7 | exbii 1848 | . . . 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 1848 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
| 11 | elvv 5760 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
| 12 | n0 4353 | . 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 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 {csn 4626 〈cop 4632 class class class wbr 5143 × cxp 5683 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 |
| This theorem is referenced by: rnsnn0 6228 dmsn0 6229 dmsn0el 6231 relsn2 6232 1stnpr 8018 1st2val 8042 mpoxopxnop0 8240 cnvfi 9216 hashfun 14476 fineqvac 35111 |
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