Theorem elsigass 34381

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 34380	. . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
2		sigasspw 34372	. . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → 𝑆 ⊆ 𝒫 ∪ 𝑆)
3	1, 2	syl 17	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ 𝒫 ∪ 𝑆)
4	3	sselda 3936	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 ∪ 𝑆)
5	4	elpwid 4563	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141   ⊆ wss 3904  𝒫 cpw 4554  ∪ cuni 4864  ran crn 5646  ‘cfv 6515  sigAlgebracsiga 34364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6471  df-fun 6517  df-fn 6518  df-fv 6523  df-siga 34365

This theorem is referenced by: (None)