Theorem unab 2270

Description: Union of two class abstractions.

Assertion

Ref	Expression
unab	|- ({x | ph} u. {x | ps}) = {x | (ph \/ ps)}

Proof of Theorem unab

Step	Hyp	Ref	Expression
1		df-clab 1467	. . . . 5 |- (y e. {x | ph} <-> [y / x]ph)
2		df-clab 1467	. . . . 5 |- (y e. {x | ps} <-> [y / x]ps)
3	1, 2	orbi12i 257	. . . 4 |- ((y e. {x | ph} \/ y e. {x | ps}) <-> ([y / x]ph \/ [y / x]ps))
4		sbor 1237	. . . 4 |- ([y / x](ph \/ ps) <-> ([y / x]ph \/ [y / x]ps))
5	3, 4	bitr4 176	. . 3 |- ((y e. {x | ph} \/ y e. {x | ps}) <-> [y / x](ph \/ ps))
6		elun 2176	. . 3 |- (y e. ({x | ph} u. {x | ps}) <-> (y e. {x | ph} \/ y e. {x | ps}))
7		df-clab 1467	. . 3 |- (y e. {x | (ph \/ ps)} <-> [y / x](ph \/ ps))
8	5, 6, 7	3bitr4 183	. 2 |- (y e. ({x | ph} u. {x | ps}) <-> y e. {x | (ph \/ ps)})
9	8	eqriv 1477	1 |- ({x | ph} u. {x | ps}) = {x | (ph \/ ps)}

Syntax hints:   \/ wo 222   = wceq 958   e. wcel 960  [wsbc 1172  {cab 1466   u. cun 2048

This theorem is referenced by:  unrab 2273  iunun 2618  unopab 2684  oarec 4202  infxpidmlem9 7561

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-12 970  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053

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