Theorem r1ord3g

Description: Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of BellMachover p. 478. (Contributed by NM, 22-Sep-2003)

	Ref	Expression
Assertion	r1ord3g	⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 ⊆ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		r1dmlim	⊢ Lim dom 𝑅1
2		limord	⊢ ( Lim dom 𝑅1 → Ord dom 𝑅1 )
3		ordsson	⊢ ( Ord dom 𝑅1 → dom 𝑅1 ⊆ On )
4	1 2 3	mp2b	⊢ dom 𝑅1 ⊆ On
5	4	sseli	⊢ ( 𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On )
6	4	sseli	⊢ ( 𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On )
7		onsseleq	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
8	5 6 7	syl2an	⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) ) )
9		r1tr	⊢ Tr ( 𝑅1 ‘ 𝐵 )
10		r1ordg	⊢ ( 𝐵 ∈ dom 𝑅1 → ( 𝐴 ∈ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) )
11	10	adantl	⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 ∈ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) )
12		trss	⊢ ( Tr ( 𝑅1 ‘ 𝐵 ) → ( ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) → ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐵 ) ) )
13	9 11 12	mpsylsyld	⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 ∈ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐵 ) ) )
14		fveq2	⊢ ( 𝐴 = 𝐵 → ( 𝑅1 ‘ 𝐴 ) = ( 𝑅1 ‘ 𝐵 ) )
15		eqimss	⊢ ( ( 𝑅1 ‘ 𝐴 ) = ( 𝑅1 ‘ 𝐵 ) → ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐵 ) )
16	14 15	syl	⊢ ( 𝐴 = 𝐵 → ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐵 ) )
17	16	a1i	⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 = 𝐵 → ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐵 ) ) )
18	13 17	jaod	⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → ( ( 𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ) → ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐵 ) ) )
19	8 18	sylbid	⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 ⊆ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ⊆ ( 𝑅1 ‘ 𝐵 ) ) )