Theorem n0el2 38984

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 5909	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = 𝐴)
2		n0el 4319	. 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥)
3		cnvepres 38953	. . . 4 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
4	3	dmeqi 5894	. . 3 ⊢ dom (◡ E ↾ 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
5	4	eqeq1i 2768	. 2 ⊢ (dom (◡ E ↾ 𝐴) = 𝐴 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = 𝐴)
6	1, 2, 5	3bitr4i 306	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∅c0 4286  {copab 5173   E cep 5560  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-res 5673

This theorem is referenced by:  n0elim  39384  membpartlem19  39563