Theorem wunsuc 10670

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 6338	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2, 3	wunsn 10669	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
5	2, 3, 4	wunun 10663	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
6	1, 5	eqeltrid 2832	1 ⊢ (𝜑 → suc 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2109   ∪ cun 3912  {csn 4589  suc csuc 6334  WUnicwun 10653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-v 3449  df-un 3919  df-ss 3931  df-sn 4590  df-pr 4592  df-uni 4872  df-tr 5215  df-suc 6338  df-wun 10655

This theorem is referenced by: (None)