Theorem 0er 6831

Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)

Assertion

Ref	Expression
0er	|- (/) Er (/)

Proof of Theorem 0er

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rel0 4897	. . . 4 |- Rel (/)
2	1	a1i 9	. . 3 |- ( T. -> Rel (/) )
3		df-br 4126	. . . . 5 |- ( x (/) y <-> <. x , y >. e. (/) )
4		noel 3525	. . . . . 6 |- -. <. x , y >. e. (/)
5	4	pm2.21i 655	. . . . 5 |- ( <. x , y >. e. (/) -> y (/) x )
6	3, 5	sylbi 121	. . . 4 |- ( x (/) y -> y (/) x )
7	6	adantl 277	. . 3 |- ( ( T. /\ x (/) y ) -> y (/) x )
8	4	pm2.21i 655	. . . . 5 |- ( <. x , y >. e. (/) -> x (/) z )
9	3, 8	sylbi 121	. . . 4 |- ( x (/) y -> x (/) z )
10	9	ad2antrl 494	. . 3 |- ( ( T. /\ ( x (/) y /\ y (/) z ) ) -> x (/) z )
11		noel 3525	. . . . . 6 |- -. x e. (/)
12		noel 3525	. . . . . 6 |- -. <. x , x >. e. (/)
13	11, 12	2false 713	. . . . 5 |- ( x e. (/) <-> <. x , x >. e. (/) )
14		df-br 4126	. . . . 5 |- ( x (/) x <-> <. x , x >. e. (/) )
15	13, 14	bitr4i 187	. . . 4 |- ( x e. (/) <-> x (/) x )
16	15	a1i 9	. . 3 |- ( T. -> ( x e. (/) <-> x (/) x ) )
17	2, 7, 10, 16	iserd 6823	. 2 |- ( T. -> (/) Er (/) )
18	17	mptru 1411	1 |- (/) Er (/)

Syntax hints:   <-> wb 105   T. wtru 1403   e. wcel 2209   (/)c0 3520   <.cop 3708   class class class wbr 4125   Rel wrel 4774   Er wer 6794

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-er 6797

This theorem is referenced by: (None)