Theorem prtlem400 34155

Description: Lemma for prter2 34166 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem13.1	|- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }

Assertion

Ref	Expression
prtlem400	|- -. (/) e. ( U. A /. .~ )

Distinct variable group:   x, u, y, A

Allowed substitution hints:   .~ ( x, y, u)

Proof of Theorem prtlem400

Step	Hyp	Ref	Expression
1		neirr 2803	. 2 |- -. (/) =/= (/)
2		prtlem13.1	. . . 4 |- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
3	2	prtlem16 34154	. . 3 |- dom .~ = U. A
4		elqsn0 7816	. . 3 |- ( ( dom .~ = U. A /\ (/) e. ( U. A /. .~ ) ) -> (/) =/= (/) )
5	3, 4	mpan 706	. 2 |- ( (/) e. ( U. A /. .~ ) -> (/) =/= (/) )
6	1, 5	mto 188	1 |- -. (/) e. ( U. A /. .~ )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   (/)c0 3915   U.cuni 4436   {copab 4712   dom cdm 5114   /.cqs 7741

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  prter2  34166