Theorem inposetlem 10475

Description: Lemma for inposet 10477. Definition of inclusion of sets using a class. (Part of FL's sandbox.)

Hypotheses

Ref	Expression
inposetlem.1	|- A e. V
inposetlem.2	|- B e. V
inposetlem.3	|- C = {<.x, y>. | x (_ y}

Assertion

Ref	Expression
inposetlem	|- (ACB <-> A (_ B)

Distinct variable groups:   x,A,y   x,B,y

Proof of Theorem inposetlem

Step	Hyp	Ref	Expression
1		inposetlem.1	. 2 |- A e. V
2		inposetlem.2	. 2 |- B e. V
3		sseq1 2085	. 2 |- (x = A -> (x (_ y <-> A (_ y))
4		sseq2 2086	. 2 |- (y = B -> (A (_ y <-> A (_ B))
5		inposetlem.3	. 2 |- C = {<.x, y>. | x (_ y}
6	1, 2, 3, 4, 5	brab 2827	1 |- (ACB <-> A (_ B)

Syntax hints:   <-> wb 146   = wceq 958   e. wcel 960  Vcvv 1814   (_ wss 2050   class class class wbr 2624  {copab 2671

This theorem is referenced by:  inposet 10477

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672

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