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Theorem rankelb 8725
 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 8706 . . . . . 6 (𝐵 (𝑅1 “ On) → 𝐵 (𝑅1 “ On))
21sseld 3635 . . . . 5 (𝐵 (𝑅1 “ On) → (𝐴𝐵𝐴 (𝑅1 “ On)))
3 rankidn 8723 . . . . 5 (𝐴 (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
42, 3syl6 35 . . . 4 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
54imp 444 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
6 rankon 8696 . . . . 5 (rank‘𝐵) ∈ On
7 rankon 8696 . . . . 5 (rank‘𝐴) ∈ On
8 ontri1 5795 . . . . 5 (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)))
96, 7, 8mp2an 708 . . . 4 ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))
10 rankdmr1 8702 . . . . . 6 (rank‘𝐵) ∈ dom 𝑅1
11 rankdmr1 8702 . . . . . 6 (rank‘𝐴) ∈ dom 𝑅1
12 r1ord3g 8680 . . . . . 6 (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))))
1310, 11, 12mp2an 708 . . . . 5 ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))
14 r1rankidb 8705 . . . . . . 7 (𝐵 (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
1514sselda 3636 . . . . . 6 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐵)))
16 ssel 3630 . . . . . 6 ((𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)) → (𝐴 ∈ (𝑅1‘(rank‘𝐵)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
1715, 16syl5com 31 . . . . 5 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → ((𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
1813, 17syl5 34 . . . 4 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → ((rank‘𝐵) ⊆ (rank‘𝐴) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
199, 18syl5bir 233 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (¬ (rank‘𝐴) ∈ (rank‘𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
205, 19mt3d 140 . 2 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
2120ex 449 1 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2030   ⊆ wss 3607  ∪ cuni 4468  dom cdm 5143   “ cima 5146  Oncon0 5761  ‘cfv 5926  𝑅1cr1 8663  rankcrnk 8664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666 This theorem is referenced by:  wfelirr  8726  rankval3b  8727  rankel  8740  rankunb  8751  rankuni2b  8754  rankcf  9637
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