Theorem wunsuc 10702

Description: A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wunsuc	⊢ (𝜑 → suc 𝐴 ∈ 𝑈)

Proof of Theorem wunsuc

Step	Hyp	Ref	Expression
1		df-suc 6367	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2, 3	wunsn 10701	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
5	2, 3, 4	wunun 10695	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
6	1, 5	eqeltrid 2873	1 ⊢ (𝜑 → suc 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2149   ∪ cun 3911  {csn 4594  suc csuc 6363  WUnicwun 10685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-v 3465  df-un 3918  df-ss 3930  df-sn 4595  df-pr 4597  df-uni 4877  df-tr 5223  df-suc 6367  df-wun 10687

This theorem is referenced by: (None)