Theorem rab0 3243

Description: Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rab0	⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Proof of Theorem rab0

Step	Hyp	Ref	Expression
1		noel 3225	. . . . 5 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnanr 839	. . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
3		equid 1589	. . . . 5 ⊢ 𝑥 = 𝑥
4	3	notnoti 574	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
5	2, 4	2false 617	. . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
6	5	abbii 2153	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7		df-rab 2312	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8		dfnul2 3223	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2070	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  {crab 2307  ∅c0 3221

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2312  df-v 2556  df-dif 2917  df-nul 3222

This theorem is referenced by: (None)