Theorem wunsuc 10697

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 6366	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2, 3	wunsn 10696	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
5	2, 3, 4	wunun 10690	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
6	1, 5	eqeltrid 2867	1 ⊢ (𝜑 → suc 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2143   ∪ cun 3903  {csn 4589  suc csuc 6362  WUnicwun 10680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-v 3457  df-un 3910  df-ss 3922  df-sn 4590  df-pr 4592  df-uni 4873  df-tr 5219  df-suc 6366  df-wun 10682

This theorem is referenced by: (None)