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Theorem rankssb 9846

Description: The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

**Assertion**
Ref	Expression
rankssb	⊢ (𝐵 ∈ ∪ (𝑅₁ “ On) → (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))

**Proof of Theorem rankssb**
Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
2		r1rankidb 9802	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅₁ “ On) → 𝐵 ⊆ (𝑅₁‘(rank‘𝐵)))
3	2	adantr 480	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ (𝑅₁‘(rank‘𝐵)))
4	1, 3	sstrd 3992	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝑅₁‘(rank‘𝐵)))
5		sswf 9806	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ ∪ (𝑅₁ “ On))
6		rankdmr1 9799	. . . 4 ⊢ (rank‘𝐵) ∈ dom 𝑅₁
7		rankr1bg 9801	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝐵) ∈ dom 𝑅₁) → (𝐴 ⊆ (𝑅₁‘(rank‘𝐵)) ↔ (rank‘𝐴) ⊆ (rank‘𝐵)))
8	5, 6, 7	sylancl 585	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ⊆ (𝑅₁‘(rank‘𝐵)) ↔ (rank‘𝐴) ⊆ (rank‘𝐵)))
9	4, 8	mpbid 231	. 2 ⊢ ((𝐵 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ⊆ 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐵))
10	9	ex 412	1 ⊢ (𝐵 ∈ ∪ (𝑅₁ “ On) → (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 ⊆ wss 3948 ∪ cuni 4908 dom cdm 5676 “ cima 5679 Oncon0 6364 ‘cfv 6543 𝑅₁cr1 9760 rankcrnk 9761

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7859 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-r1 9762 df-rank 9763

This theorem is referenced by: rankss 9847 rankunb 9848 rankuni2b 9851 rankr1id 9860

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