Theorem unisalgen 40558

Description: The union of a set belongs to the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
unisalgen.x	|- ( ph -> X e. V )
unisalgen.s	|- S = (SalGen ` X )
unisalgen.u	|- U = U. X

Assertion

Ref	Expression
unisalgen	|- ( ph -> U e. S )

Proof of Theorem unisalgen

Step	Hyp	Ref	Expression
1		unisalgen.x	. . . 4 |- ( ph -> X e. V )
2		unisalgen.s	. . . 4 |- S = (SalGen ` X )
3		unisalgen.u	. . . 4 |- U = U. X
4	1, 2, 3	salgenuni 40555	. . 3 |- ( ph -> U. S = U )
5	4	eqcomd 2628	. 2 |- ( ph -> U = U. S )
6	2	a1i 11	. . . 4 |- ( ph -> S = (SalGen ` X ) )
7		salgencl 40550	. . . . 5 |- ( X e. V -> (SalGen ` X ) e. SAlg )
8	1, 7	syl 17	. . . 4 |- ( ph -> (SalGen ` X ) e. SAlg )
9	6, 8	eqeltrd 2701	. . 3 |- ( ph -> S e. SAlg )
10		saluni 40544	. . 3 |- ( S e. SAlg -> U. S e. S )
11	9, 10	syl 17	. 2 |- ( ph -> U. S e. S )
12	5, 11	eqeltrd 2701	1 |- ( ph -> U e. S )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   U.cuni 4436   ` cfv 5888  SAlgcsalg 40528  SalGencsalgen 40532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-salg 40529  df-salgen 40533

This theorem is referenced by:  salgensscntex  40562