Theorem prwf

Description: An unordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	prwf	⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → { 𝐴 , 𝐵 } ∈ ∪ ( 𝑅1 “ On ) )

Proof

Step	Hyp	Ref	Expression
1		df-pr	⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
2		snwf	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → { 𝐴 } ∈ ∪ ( 𝑅1 “ On ) )
3		snwf	⊢ ( 𝐵 ∈ ∪ ( 𝑅1 “ On ) → { 𝐵 } ∈ ∪ ( 𝑅1 “ On ) )
4		unwf	⊢ ( ( { 𝐴 } ∈ ∪ ( 𝑅1 “ On ) ∧ { 𝐵 } ∈ ∪ ( 𝑅1 “ On ) ) ↔ ( { 𝐴 } ∪ { 𝐵 } ) ∈ ∪ ( 𝑅1 “ On ) )
5	4	biimpi	⊢ ( ( { 𝐴 } ∈ ∪ ( 𝑅1 “ On ) ∧ { 𝐵 } ∈ ∪ ( 𝑅1 “ On ) ) → ( { 𝐴 } ∪ { 𝐵 } ) ∈ ∪ ( 𝑅1 “ On ) )
6	2 3 5	syl2an	⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( { 𝐴 } ∪ { 𝐵 } ) ∈ ∪ ( 𝑅1 “ On ) )
7	1 6	eqeltrid	⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → { 𝐴 , 𝐵 } ∈ ∪ ( 𝑅1 “ On ) )