Theorem rab0 3523

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 3498	. . . . 5 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnanr 937	. . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
3		equid 1749	. . . . 5 ⊢ 𝑥 = 𝑥
4	3	notnoti 650	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
5	2, 4	2false 708	. . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
6	5	abbii 2347	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7		df-rab 2519	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8		dfnul2 3496	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2262	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {cab 2217  {crab 2514  ∅c0 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-dif 3202  df-nul 3495

This theorem is referenced by:  ssfirab  7128  sup00  7201  vtxdgfi0e  16145  clwwlk0on0  16281