Theorem wunsuc 10626

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 6321	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2, 3	wunsn 10625	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
5	2, 3, 4	wunun 10619	. 2 ⊢ (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
6	1, 5	eqeltrid 2838	1 ⊢ (𝜑 → suc 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2113   ∪ cun 3897  {csn 4578  suc csuc 6317  WUnicwun 10609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-v 3440  df-un 3904  df-ss 3916  df-sn 4579  df-pr 4581  df-uni 4862  df-tr 5204  df-suc 6321  df-wun 10611

This theorem is referenced by: (None)