Theorem sigasspw 34273

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.

Step	Hyp	Ref	Expression
1		elex 3461	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V)
2		issiga 34269	. . . 4 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝐴) ↔ (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
3	2	biimpa 476	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
4	1, 3	mpancom 688	. 2 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
5	4	simpld 494	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863   class class class wbr 5098  ‘cfv 6492  ωcom 7808   ≼ cdom 8881  sigAlgebracsiga 34265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-siga 34266

This theorem is referenced by:  elsigass  34282  insiga  34294  sigapisys  34312  sigaldsys  34316  brsigasspwrn  34342  1stmbfm  34417  2ndmbfm  34418