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Theorem rankssb 9253
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) → (𝐴𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))

Proof of Theorem rankssb
StepHypRef Expression
1 simpr 488 . . . 4 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → 𝐴𝐵)
2 r1rankidb 9209 . . . . 5 (𝐵 (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
32adantr 484 . . . 4 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
41, 3sstrd 3953 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → 𝐴 ⊆ (𝑅1‘(rank‘𝐵)))
5 sswf 9213 . . . 4 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → 𝐴 (𝑅1 “ On))
6 rankdmr1 9206 . . . 4 (rank‘𝐵) ∈ dom 𝑅1
7 rankr1bg 9208 . . . 4 ((𝐴 (𝑅1 “ On) ∧ (rank‘𝐵) ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘(rank‘𝐵)) ↔ (rank‘𝐴) ⊆ (rank‘𝐵)))
85, 6, 7sylancl 589 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (𝐴 ⊆ (𝑅1‘(rank‘𝐵)) ↔ (rank‘𝐴) ⊆ (rank‘𝐵)))
94, 8mpbid 235 . 2 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (rank‘𝐴) ⊆ (rank‘𝐵))
109ex 416 1 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wss 3910   cuni 4811  dom cdm 5528  cima 5531  Oncon0 6164  cfv 6328  𝑅1cr1 9167  rankcrnk 9168
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-r1 9169  df-rank 9170
This theorem is referenced by:  rankss  9254  rankunb  9255  rankuni2b  9258  rankr1id  9267
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