Theorem eupth2lem1 16613

Description: Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

Assertion

Ref	Expression
eupth2lem1	|- ( U e. V -> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )

Proof of Theorem eupth2lem1

Step	Hyp	Ref	Expression
1		elif 3649	. . 3 |- ( U e. if ( A = B , (/) , { A , B } ) <-> ( ( A = B /\ U e. (/) ) \/ ( -. A = B /\ U e. { A , B } ) ) )
2		noel 3525	. . . . 5 |- -. U e. (/)
3	2	intnan 941	. . . 4 |- -. ( A = B /\ U e. (/) )
4		biorf 756	. . . 4 |- ( -. ( A = B /\ U e. (/) ) -> ( ( -. A = B /\ U e. { A , B } ) <-> ( ( A = B /\ U e. (/) ) \/ ( -. A = B /\ U e. { A , B } ) ) ) )
5	3, 4	ax-mp 5	. . 3 |- ( ( -. A = B /\ U e. { A , B } ) <-> ( ( A = B /\ U e. (/) ) \/ ( -. A = B /\ U e. { A , B } ) ) )
6	1, 5	bitr4i 187	. 2 |- ( U e. if ( A = B , (/) , { A , B } ) <-> ( -. A = B /\ U e. { A , B } ) )
7		df-ne 2421	. . . . 5 |- ( A =/= B <-> -. A = B )
8	7	bicomi 132	. . . 4 |- ( -. A = B <-> A =/= B )
9	8	a1i 9	. . 3 |- ( U e. V -> ( -. A = B <-> A =/= B ) )
10		elprg 3725	. . 3 |- ( U e. V -> ( U e. { A , B } <-> ( U = A \/ U = B ) ) )
11	9, 10	anbi12d 477	. 2 |- ( U e. V -> ( ( -. A = B /\ U e. { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )
12	6, 11	bitrid 192	1 |- ( U e. V -> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   (/)c0 3520   ifcif 3635   {cpr 3706

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-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-if 3636  df-sn 3711  df-pr 3712

This theorem is referenced by:  eupth2lem2dc  16614  eupth2lem3lem6fi  16626