Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dm0 | Structured version Visualization version GIF version |
Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dm0 | ⊢ dom ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3888 | . 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅) | |
2 | noel 3878 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
3 | 2 | nex 1722 | . . 3 ⊢ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅ |
4 | vex 3176 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 4 | eldm2 5244 | . . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅) |
6 | 3, 5 | mtbir 312 | . 2 ⊢ ¬ 𝑥 ∈ dom ∅ |
7 | 1, 6 | mpgbir 1717 | 1 ⊢ dom ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∅c0 3874 〈cop 4131 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 |
This theorem is referenced by: dmxpid 5266 rn0 5298 dmxpss 5484 fn0 5924 f0dom0 6002 f10d 6082 f1o00 6083 0fv 6137 1stval 7061 bropopvvv 7142 bropfvvvv 7144 supp0 7187 tz7.44lem1 7388 tz7.44-2 7390 tz7.44-3 7391 oicl 8317 oif 8318 swrd0 13286 dmtrclfv 13607 strlemor0 15795 symgsssg 17710 symgfisg 17711 psgnunilem5 17737 dvbsss 23472 perfdvf 23473 uhgr0e 25737 uhgr0 25739 uhgra0 25838 umgra0 25854 clwwlknprop 26300 2wlkonot3v 26402 2spthonot3v 26403 eupa0 26501 dmadjrnb 28149 f1ocnt 28946 mbfmcst 29648 0rrv 29840 matunitlindf 32577 ismgmOLD 32819 conrel2d 36975 neicvgbex 37430 iblempty 38857 usgr0 40469 egrsubgr 40501 0grsubgr 40502 vtxdg0e 40689 eupth0 41382 |
Copyright terms: Public domain | W3C validator |