Theorem rab0OLD 3956

Description: Obsolete proof of rab0 3955 as of 14-Jul-2021. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rab0OLD	|- { x e. (/) | ph } = (/)

Proof of Theorem rab0OLD

Step	Hyp	Ref	Expression
1		equid 1939	. . . . 5 |- x = x
2		noel 3919	. . . . . 6 |- -. x e. (/)
3	2	intnanr 961	. . . . 5 |- -. ( x e. (/) /\ ph )
4	1, 3	2th 254	. . . 4 |- ( x = x <-> -. ( x e. (/) /\ ph ) )
5	4	con2bii 347	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2739	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2921	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3917	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2654	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)