Theorem gruelss 10709

Description: A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)

Assertion

Ref	Expression
gruelss	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)

Proof of Theorem gruelss

Step	Hyp	Ref	Expression
1		grutr 10708	. 2 ⊢ (𝑈 ∈ Univ → Tr 𝑈)
2		trss 5190	. . 3 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈))
3	2	imp 407	. 2 ⊢ ((Tr 𝑈 ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)
4	1, 3	sylan 586	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119   ⊆ wss 3883  Tr wtr 5180  Univcgru 10705

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-ext 2711

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-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-tr 5181  df-iota 6442  df-fv 6494  df-ov 7360  df-gru 10706

This theorem is referenced by:  gruss  10711  gruuni  10715  gruel  10718  grur1a  10734  grur1  10735