Theorem gruelss 10688

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 10687	. 2 ⊢ (𝑈 ∈ Univ → Tr 𝑈)
2		trss 5209	. . 3 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈))
3	2	imp 406	. 2 ⊢ ((Tr 𝑈 ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)
4	1, 3	sylan 580	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ⊆ wss 3903  Tr wtr 5199  Univcgru 10684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-tr 5200  df-iota 6438  df-fv 6490  df-ov 7352  df-gru 10685

This theorem is referenced by:  gruss  10690  gruuni  10694  gruel  10697  grur1a  10713  grur1  10714