Theorem gruel 10901

Description: Any element of an element of a Grothendieck universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
gruel	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝑈)

Proof of Theorem gruel

Step	Hyp	Ref	Expression
1		gruelss 10892	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)
2	1	sseld 3912	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝑈))
3	2	3impia 1135	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146  Univcgru 10888

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-tr 5216  df-iota 6501  df-fv 6555  df-ov 7433  df-gru 10889

This theorem is used by:  gruf  10909  grumnudlem  45412