Theorem grutr 10716

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 10715	. . 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 267	. 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 494	1 ⊢ (𝑈 ∈ Univ → Tr 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114  ∀wral 3052  𝒫 cpw 4556  {cpr 4584  ∪ cuni 4865  Tr wtr 5207  ran crn 5633  (class class class)co 7368   ↑_m cmap 8775  Univcgru 10713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-tr 5208  df-iota 6456  df-fv 6508  df-ov 7371  df-gru 10714

This theorem is referenced by:  gruelss  10717  gruwun  10736  intgru  10737  gruina  10741  grur1  10743  grutsk  10745