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Theorem sigasspw 30409
Description: A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)
Assertion
Ref Expression
sigasspw (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴)

Proof of Theorem sigasspw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3316 . . 3 (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V)
2 issiga 30404 . . . 4 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝐴) ↔ (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32biimpa 502 . . 3 ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
41, 3mpancom 706 . 2 (𝑆 ∈ (sigAlgebra‘𝐴) → (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴𝑆 ∧ ∀𝑥𝑆 (𝐴𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
54simpld 477 1 (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wcel 2103  wral 3014  Vcvv 3304  cdif 3677  wss 3680  𝒫 cpw 4266   cuni 4544   class class class wbr 4760  cfv 6001  ωcom 7182  cdom 8070  sigAlgebracsiga 30400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-siga 30401
This theorem is referenced by:  elsigass  30418  insiga  30430  sigapisys  30448  sigaldsys  30452  brsigasspwrn  30478  1stmbfm  30552  2ndmbfm  30553
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