Theorem wunelss 9530

Description: The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	|- ( ph -> U e. WUni )
wununi.2	|- ( ph -> A e. U )

Assertion

Ref	Expression
wunelss	|- ( ph -> A C_ U )

Proof of Theorem wunelss

Step	Hyp	Ref	Expression
1		wununi.1	. . 3 |- ( ph -> U e. WUni )
2		wuntr 9527	. . 3 |- ( U e. WUni -> Tr U )
3	1, 2	syl 17	. 2 |- ( ph -> Tr U )
4		wununi.2	. 2 |- ( ph -> A e. U )
5		trss 4761	. 2 |- ( Tr U -> ( A e. U -> A C_ U ) )
6	3, 4, 5	sylc 65	1 |- ( ph -> A C_ U )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   Tr wtr 4752  WUnicwun 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753  df-wun 9524

This theorem is referenced by:  wunss  9534  wunf  9549  wuncval2  9569