Theorem rab0 3437

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 3413	. . . . 5 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnanr 920	. . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
3		equid 1689	. . . . 5 ⊢ 𝑥 = 𝑥
4	3	notnoti 635	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
5	2, 4	2false 691	. . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
6	5	abbii 2282	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7		df-rab 2453	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8		dfnul2 3411	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2196	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1343   ∈ wcel 2136  {cab 2151  {crab 2448  ∅c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by:  ssfirab  6899  sup00  6968