Theorem grutr 10550

Description: A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
grutr	⊢ (𝑈 ∈ Univ → Tr 𝑈)

Proof of Theorem grutr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elgrug 10549	. . 3 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))
2	1	ibi 266	. 2 ⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)))
3	2	simpld 495	1 ⊢ (𝑈 ∈ Univ → Tr 𝑈)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   ∈ wcel 2110  ∀wral 3066  𝒫 cpw 4539  {cpr 4569  ∪ cuni 4845  Tr wtr 5196  ran crn 5591  (class class class)co 7271   ↑_m cmap 8598  Univcgru 10547

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-tr 5197  df-iota 6390  df-fv 6440  df-ov 7274  df-gru 10548

This theorem is referenced by:  gruelss  10551  gruwun  10570  intgru  10571  gruina  10575  grur1  10577  grutsk  10579