Theorem rab0 3475

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 3450	. . . . 5 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnanr 931	. . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑)
3		equid 1712	. . . . 5 ⊢ 𝑥 = 𝑥
4	3	notnoti 646	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
5	2, 4	2false 702	. . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥)
6	5	abbii 2309	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7		df-rab 2481	. 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)}
8		dfnul2 3448	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2224	1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  {crab 2476  ∅c0 3446

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-dif 3155  df-nul 3447

This theorem is referenced by:  ssfirab  6990  sup00  7062