Theorem eupth2lem1 16453

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 3634	. . 3 |- ( U e. if ( A = B , (/) , { A , B } ) <-> ( ( A = B /\ U e. (/) ) \/ ( -. A = B /\ U e. { A , B } ) ) )
2		noel 3512	. . . . 5 |- -. U e. (/)
3	2	intnan 937	. . . 4 |- -. ( A = B /\ U e. (/) )
4		biorf 752	. . . 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 2413	. . . . 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 3709	. . 3 |- ( U e. V -> ( U e. { A , B } <-> ( U = A \/ U = B ) ) )
11	9, 10	anbi12d 473	. 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 716   = wceq 1398   e. wcel 2203   =/= wne 2412   (/)c0 3508   ifcif 3620   {cpr 3690

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-v 2815  df-dif 3213  df-un 3215  df-nul 3509  df-if 3621  df-sn 3695  df-pr 3696

This theorem is referenced by:  eupth2lem2dc  16454  eupth2lem3lem6fi  16466