Theorem gruel 10815

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 10806	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)
2	1	sseld 3930	. 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 2145  Univcgru 10802

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

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-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-tr 5213  df-iota 6489  df-fv 6541  df-ov 7417  df-gru 10803

This theorem is used by:  gruf  10823  grumnudlem  45112