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Theorem rankelb 8547
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 8528 . . . . . 6 (𝐵 (𝑅1 “ On) → 𝐵 (𝑅1 “ On))
21sseld 3566 . . . . 5 (𝐵 (𝑅1 “ On) → (𝐴𝐵𝐴 (𝑅1 “ On)))
3 rankidn 8545 . . . . 5 (𝐴 (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
42, 3syl6 34 . . . 4 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
54imp 443 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
6 rankon 8518 . . . . 5 (rank‘𝐵) ∈ On
7 rankon 8518 . . . . 5 (rank‘𝐴) ∈ On
8 ontri1 5660 . . . . 5 (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)))
96, 7, 8mp2an 703 . . . 4 ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))
10 rankdmr1 8524 . . . . . 6 (rank‘𝐵) ∈ dom 𝑅1
11 rankdmr1 8524 . . . . . 6 (rank‘𝐴) ∈ dom 𝑅1
12 r1ord3g 8502 . . . . . 6 (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))))
1310, 11, 12mp2an 703 . . . . 5 ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))
14 r1rankidb 8527 . . . . . . 7 (𝐵 (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
1514sselda 3567 . . . . . 6 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐵)))
16 ssel 3561 . . . . . 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 33 . . . 4 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → ((rank‘𝐵) ⊆ (rank‘𝐴) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
199, 18syl5bir 231 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (¬ (rank‘𝐴) ∈ (rank‘𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
205, 19mt3d 138 . 2 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
2120ex 448 1 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wcel 1976  wss 3539   cuni 4366  dom cdm 5028  cima 5031  Oncon0 5626  cfv 5790  𝑅1cr1 8485  rankcrnk 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-r1 8487  df-rank 8488
This theorem is referenced by:  wfelirr  8548  rankval3b  8549  rankel  8562  rankunb  8573  rankuni2b  8576  rankcf  9455
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