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Theorem rankelb 9241
 Description: The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankelb (𝐵 (𝑅1 “ On) → (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))

Proof of Theorem rankelb
StepHypRef Expression
1 r1elssi 9222 . . . . . 6 (𝐵 (𝑅1 “ On) → 𝐵 (𝑅1 “ On))
21sseld 3917 . . . . 5 (𝐵 (𝑅1 “ On) → (𝐴𝐵𝐴 (𝑅1 “ On)))
3 rankidn 9239 . . . . 5 (𝐴 (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
42, 3syl6 35 . . . 4 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
54imp 410 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
6 rankon 9212 . . . . 5 (rank‘𝐵) ∈ On
7 rankon 9212 . . . . 5 (rank‘𝐴) ∈ On
8 ontri1 6197 . . . . 5 (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)))
96, 7, 8mp2an 691 . . . 4 ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))
10 rankdmr1 9218 . . . . . 6 (rank‘𝐵) ∈ dom 𝑅1
11 rankdmr1 9218 . . . . . 6 (rank‘𝐴) ∈ dom 𝑅1
12 r1ord3g 9196 . . . . . 6 (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))))
1310, 11, 12mp2an 691 . . . . 5 ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))
14 r1rankidb 9221 . . . . . 6 (𝐵 (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
1514sselda 3918 . . . . 5 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐵)))
16 ssel 3911 . . . . 5 ((𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)) → (𝐴 ∈ (𝑅1‘(rank‘𝐵)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
1713, 15, 16syl2imc 41 . . . 4 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → ((rank‘𝐵) ⊆ (rank‘𝐴) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
189, 17syl5bir 246 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (¬ (rank‘𝐴) ∈ (rank‘𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
195, 18mt3d 150 . 2 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
2019ex 416 1 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112   ⊆ wss 3884  ∪ cuni 4803  dom cdm 5523   “ cima 5526  Oncon0 6163  ‘cfv 6328  𝑅1cr1 9179  rankcrnk 9180 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  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 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181  df-rank 9182 This theorem is referenced by:  wfelirr  9242  rankval3b  9243  rankel  9256  rankunb  9267  rankuni2b  9270  rankcf  10192
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