Theorem sigasspw 34484

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 3476	. . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V)
2		issiga 34480	. . . 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 481	. . 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 700	. 2 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
5	4	simpld 499	1 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  𝒫 cpw 4563  ∪ cuni 4873   class class class wbr 5110  ‘cfv 6538  ωcom 7863   ≼ cdom 8942  sigAlgebracsiga 34476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-siga 34477

This theorem is referenced by:  elsigass  34493  insiga  34505  sigapisys  34523  sigaldsys  34527  brsigasspwrn  34553  1stmbfm  34628  2ndmbfm  34629