Theorem elsigass 34840

Description: An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)

Assertion

Ref	Expression
elsigass	⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)

Proof of Theorem elsigass

Step	Hyp	Ref	Expression
1		sgon 34839	. . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
2		sigasspw 34831	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → 𝑆 ⊆ 𝒫 ∪ 𝑆)
3	1, 2	syl 18	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ 𝒫 ∪ 𝑆)
4	3	sselda 3913	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 ∪ 𝑆)
5	4	elpwid 4569	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4870  ran crn 5657  ‘cfv 6561  sigAlgebracsiga 34823

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-fv 6569  df-siga 34824

This theorem is used by:  difelsiga  34850