Theorem elsigass 34316

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 34315	. . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
2		sigasspw 34307	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → 𝑆 ⊆ 𝒫 ∪ 𝑆)
3	1, 2	syl 17	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ 𝒫 ∪ 𝑆)
4	3	sselda 3922	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 ∪ 𝑆)
5	4	elpwid 4545	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119   ⊆ wss 3890  𝒫 cpw 4536  ∪ cuni 4845  ran crn 5626  ‘cfv 6492  sigAlgebracsiga 34299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-siga 34300

This theorem is referenced by: (None)