Theorem dford4lem2 13860

Description: Lemma for dford4 . A new-style ordinal is irreflexive in membership.

Assertion

Ref	Expression
dford4lem2	|- (A = {x | (x C. A /\ Tr x)} -> -. A e. A)

Distinct variable group:   x,A

Proof of Theorem dford4lem2

Step	Hyp	Ref	Expression
1		eleq2 1958	. . 3 |- (A = {x | (x C. A /\ Tr x)} -> (A e. A <-> A e. {x | (x C. A /\ Tr x)}))
2		psseq1 2697	. . . . . . 7 |- (x = A -> (x C. A <-> A C. A))
3		treq 3417	. . . . . . 7 |- (x = A -> (Tr x <-> Tr A))
4	2, 3	anbi12d 690	. . . . . 6 |- (x = A -> ((x C. A /\ Tr x) <-> (A C. A /\ Tr A)))
5	4	elabg 2405	. . . . 5 |- (A e. {x | (x C. A /\ Tr x)} -> (A e. {x | (x C. A /\ Tr x)} <-> (A C. A /\ Tr A)))
6	5	ibi 652	. . . 4 |- (A e. {x | (x C. A /\ Tr x)} -> (A C. A /\ Tr A))
7		pssirr 2708	. . . . . 6 |- -. A C. A
8	7	pm2.21i 93	. . . . 5 |- (A C. A -> -. A e. A)
9	8	adantr 425	. . . 4 |- ((A C. A /\ Tr A) -> -. A e. A)
10	6, 9	syl 12	. . 3 |- (A e. {x | (x C. A /\ Tr x)} -> -. A e. A)
11	1, 10	syl6bi 231	. 2 |- (A = {x | (x C. A /\ Tr x)} -> (A e. A -> -. A e. A))
12	11	pm2.01d 105	1 |- (A = {x | (x C. A /\ Tr x)} -> -. A e. A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   C. wpss 2594  Tr wtr 3411

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-pss 2607  df-uni 3178  df-tr 3412

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