Theorem rankssb 9537

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 9493	. . . . 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 9497	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
6		rankdmr1 9490	. . . 4 ⊢ (rank‘𝐵) ∈ dom 𝑅1
7		rankr1bg 9492	. . . 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)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108   ⊆ wss 3883  ∪ cuni 4836  dom cdm 5580   “ cima 5583  Oncon0 6251  ‘cfv 6418  𝑅₁cr1 9451  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453  df-rank 9454

This theorem is referenced by:  rankss  9538  rankunb  9539  rankuni2b  9542  rankr1id  9551