Theorem rankssb 9761

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 9717	. . . . 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 3927	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝑅1‘(rank‘𝐵)))
5		sswf 9721	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
6		rankdmr1 9714	. . . 4 ⊢ (rank‘𝐵) ∈ dom 𝑅1
7		rankr1bg 9716	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐵) ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘(rank‘𝐵)) ↔ (rank‘𝐴) ⊆ (rank‘𝐵)))
8	5, 6, 7	sylancl 587	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ⊆ (𝑅1‘(rank‘𝐵)) ↔ (rank‘𝐴) ⊆ (rank‘𝐵)))
9	4, 8	mpbid 232	. 2 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐵))
10	9	ex 412	1 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   ⊆ wss 3885  ∪ cuni 4840  dom cdm 5620   “ cima 5623  Oncon0 6312  ‘cfv 6487  𝑅₁cr1 9675  rankcrnk 9676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-r1 9677  df-rank 9678

This theorem is referenced by:  rankss  9762  rankunb  9763  rankuni2b  9766  rankr1id  9775  rankval4b  35231