Theorem wunsuc 10473

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 6272	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2, 3	wunsn 10472	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
5	2, 3, 4	wunun 10466	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
6	1, 5	eqeltrid 2843	1 ⊢ (𝜑 → suc 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2106   ∪ cun 3885  {csn 4561  suc csuc 6268  WUnicwun 10456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840  df-tr 5192  df-suc 6272  df-wun 10458

This theorem is referenced by: (None)