Theorem rankelun

Description: Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006) (Proof shortened by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypotheses	rankelun.1	⊢ 𝐴 ∈ V
	rankelun.2	⊢ 𝐵 ∈ V
	rankelun.3	⊢ 𝐶 ∈ V
	rankelun.4	⊢ 𝐷 ∈ V
Assertion	rankelun	⊢ ( ( ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐶 ) ∧ ( rank ‘ 𝐵 ) ∈ ( rank ‘ 𝐷 ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ( rank ‘ ( 𝐶 ∪ 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rankelun.1	⊢ 𝐴 ∈ V
2		rankelun.2	⊢ 𝐵 ∈ V
3		rankelun.3	⊢ 𝐶 ∈ V
4		rankelun.4	⊢ 𝐷 ∈ V
5		rankon	⊢ ( rank ‘ 𝐶 ) ∈ On
6		rankon	⊢ ( rank ‘ 𝐷 ) ∈ On
7	5 6	onun2i	⊢ ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) ) ∈ On
8	7	onordi	⊢ Ord ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) )
9		elun1	⊢ ( ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐶 ) → ( rank ‘ 𝐴 ) ∈ ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) ) )
10		elun2	⊢ ( ( rank ‘ 𝐵 ) ∈ ( rank ‘ 𝐷 ) → ( rank ‘ 𝐵 ) ∈ ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) ) )
11		ordunel	⊢ ( ( Ord ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) ) ∧ ( rank ‘ 𝐴 ) ∈ ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) ) ∧ ( rank ‘ 𝐵 ) ∈ ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) ) ) → ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ∈ ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) ) )
12	8 9 10 11	mp3an3an	⊢ ( ( ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐶 ) ∧ ( rank ‘ 𝐵 ) ∈ ( rank ‘ 𝐷 ) ) → ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ∈ ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) ) )
13	1 2	rankun	⊢ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) )
14	3 4	rankun	⊢ ( rank ‘ ( 𝐶 ∪ 𝐷 ) ) = ( ( rank ‘ 𝐶 ) ∪ ( rank ‘ 𝐷 ) )
15	12 13 14	3eltr4g	⊢ ( ( ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐶 ) ∧ ( rank ‘ 𝐵 ) ∈ ( rank ‘ 𝐷 ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ∈ ( rank ‘ ( 𝐶 ∪ 𝐷 ) ) )