Theorem dmsn0el 5206

Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)

Assertion

Ref	Expression
dmsn0el	⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Proof of Theorem dmsn0el

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelelxp 4754	. . . . 5 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
2	1	con2i 632	. . . 4 ⊢ (∅ ∈ 𝐴 → ¬ 𝐴 ∈ (V × V))
3		dmsnm 5202	. . . 4 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
4	2, 3	sylnib 682	. . 3 ⊢ (∅ ∈ 𝐴 → ¬ ∃𝑥 𝑥 ∈ dom {𝐴})
5		alnex 1547	. . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ dom {𝐴})
6	4, 5	sylibr 134	. 2 ⊢ (∅ ∈ 𝐴 → ∀𝑥 ¬ 𝑥 ∈ dom {𝐴})
7		eq0 3513	. 2 ⊢ (dom {𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {𝐴})
8	6, 7	sylibr 134	1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802  ∅c0 3494  {csn 3669   × cxp 4723  dom cdm 4725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-dm 4735

This theorem is referenced by: (None)