Theorem n0el2 38795

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 5891	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = 𝐴)
2		n0el 4314	. 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥)
3		cnvepres 38764	. . . 4 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
4	3	dmeqi 5876	. . 3 ⊢ dom (◡ E ↾ 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
5	4	eqeq1i 2766	. 2 ⊢ (dom (◡ E ↾ 𝐴) = 𝐴 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = 𝐴)
6	1, 2, 5	3bitr4i 305	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  ∅c0 4283  {copab 5159   E cep 5542  ^◡ccnv 5642  dom cdm 5643   ↾ cres 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-eprel 5543  df-xp 5649  df-rel 5650  df-cnv 5651  df-dm 5653  df-res 5655

This theorem is referenced by:  n0elim  39195  membpartlem19  39374