Theorem unss 1632

Description: The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse.

Assertion

Ref	Expression
unss	|- ((A (_ C /\ B (_ C) <-> (A u. B) (_ C)

Proof of Theorem unss

Step	Hyp	Ref	Expression
1		elun 1601	. . . . 5 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
2	1	imbi1i 161	. . . 4 |- ((x e. (A u. B) -> x e. C) <-> ((x e. A \/ x e. B) -> x e. C))
3		jaob 328	. . . 4 |- (((x e. A \/ x e. B) -> x e. C) <-> ((x e. A -> x e. C) /\ (x e. B -> x e. C)))
4	2, 3	bitr 151	. . 3 |- ((x e. (A u. B) -> x e. C) <-> ((x e. A -> x e. C) /\ (x e. B -> x e. C)))
5	4	bial 695	. 2 |- (A.x(x e. (A u. B) -> x e. C) <-> A.x((x e. A -> x e. C) /\ (x e. B -> x e. C)))
6		dfss2 1497	. 2 |- ((A u. B) (_ C <-> A.x(x e. (A u. B) -> x e. C))
7		dfss2 1497	. . . 4 |- (A (_ C <-> A.x(x e. A -> x e. C))
8		dfss2 1497	. . . 4 |- (B (_ C <-> A.x(x e. B -> x e. C))
9	7, 8	anbi12i 369	. . 3 |- ((A (_ C /\ B (_ C) <-> (A.x(x e. A -> x e. C) /\ A.x(x e. B -> x e. C)))
10		19.26 749	. . 3 |- (A.x((x e. A -> x e. C) /\ (x e. B -> x e. C)) <-> (A.x(x e. A -> x e. C) /\ A.x(x e. B -> x e. C)))
11	9, 10	bitr4 154	. 2 |- ((A (_ C /\ B (_ C) <-> A.x((x e. A -> x e. C) /\ (x e. B -> x e. C)))
12	5, 6, 11	3bitr4r 159	1 |- ((A (_ C /\ B (_ C) <-> (A u. B) (_ C)

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196  A.wal 672   e. wcel 1092   u. cun 1485   (_ wss 1487

This theorem is referenced by:  unssi 1633  nsspssun 1666  suceloni 2314  xpex 2488  relun 2490  er2 3201  trcl 3489  infxpidmlem11 4943  dfchj2 5325  sshjclt 5328  shlub 5347  spanun 5450

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-v 1349  df-un 1490  df-in 1491  df-ss 1492

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