Theorem rab0 3451

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 e. (/) | ph } = (/)

Proof of Theorem rab0

Step	Hyp	Ref	Expression
1		noel 3426	. . . . 5 |- -. x e. (/)
2	1	intnanr 930	. . . 4 |- -. ( x e. (/) /\ ph )
3		equid 1701	. . . . 5 |- x = x
4	3	notnoti 645	. . . 4 |- -. -. x = x
5	2, 4	2false 701	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2293	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2464	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3424	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2208	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   {crab 2459   (/)c0 3422

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-nul 3423

This theorem is referenced by:  ssfirab  6929  sup00  6998