Theorem n0el2 38363

Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 31-Jan-2018.)

Assertion

Ref	Expression
n0el2	⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴)

Proof of Theorem n0el2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmopab3 5854	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = 𝐴)
2		n0el 4309	. 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥)
3		cnvepres 38332	. . . 4 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
4	3	dmeqi 5839	. . 3 ⊢ dom (◡ E ↾ 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
5	4	eqeq1i 2736	. 2 ⊢ (dom (◡ E ↾ 𝐴) = 𝐴 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = 𝐴)
6	1, 2, 5	3bitr4i 303	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∅c0 4278  {copab 5148   E cep 5510  ^◡ccnv 5610  dom cdm 5611   ↾ cres 5613

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-eprel 5511  df-xp 5617  df-rel 5618  df-cnv 5619  df-dm 5621  df-res 5623

This theorem is referenced by:  n0elim  38688  membpartlem19  38849