Theorem elsigagen 34484

Description: Any element of a set is also an element of the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)

Assertion

Ref	Expression
elsigagen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (sigaGen‘𝐴))

Proof of Theorem elsigagen

Step	Hyp	Ref	Expression
1		sssigagen 34482	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))
2	1	sselda 3945	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (sigaGen‘𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  ‘cfv 6539  sigaGencsigagen 34475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-iota 6495  df-fun 6541  df-fv 6547  df-siga 34446  df-sigagen 34476

This theorem is referenced by:  cldssbrsiga  34524  dya2iocbrsiga  34612  dya2icobrsiga  34613  sxbrsigalem2  34623