Theorem ssequn1 1628

Description: A relationship between subclass and union. Theorem 26 of [Suppes] p. 27.

Assertion

Ref	Expression
ssequn1	|- (A (_ B <-> (A u. B) = B)

Proof of Theorem ssequn1

Step	Hyp	Ref	Expression
1		df-un 1490	. . 3 |- (A u. B) = {x | (x e. A \/ x e. B)}
2	1	cleq2i 1111	. 2 |- (B = (A u. B) <-> B = {x | (x e. A \/ x e. B)})
3		cleqcom 1103	. 2 |- ((A u. B) = B <-> B = (A u. B))
4		pm4.72 485	. . . 4 |- ((x e. A -> x e. B) <-> (x e. B <-> (x e. A \/ x e. B)))
5	4	bial 695	. . 3 |- (A.x(x e. A -> x e. B) <-> A.x(x e. B <-> (x e. A \/ x e. B)))
6		dfss2 1497	. . 3 |- (A (_ B <-> A.x(x e. A -> x e. B))
7		cleqab 1174	. . 3 |- (B = {x | (x e. A \/ x e. B)} <-> A.x(x e. B <-> (x e. A \/ x e. B)))
8	5, 6, 7	3bitr4 158	. 2 |- (A (_ B <-> B = {x | (x e. A \/ x e. B)})
9	2, 3, 8	3bitr4r 159	1 |- (A (_ B <-> (A u. B) = B)

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195  A.wal 672  {cab 1090   = wceq 1091   e. wcel 1092   u. cun 1485   (_ wss 1487

This theorem is referenced by:  ssequn2 1631  ssundif 1764  pwssun 1917  unop 1931  unisuc 2299  ordssun 2330  ordequn 2331  onuninsuc 2356  onun 2358  kmlem10 3589

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-un 1490  df-in 1491  df-ss 1492

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