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Theorem rankelb 8934
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 8915 . . . . . 6 (𝐵 (𝑅1 “ On) → 𝐵 (𝑅1 “ On))
21sseld 3797 . . . . 5 (𝐵 (𝑅1 “ On) → (𝐴𝐵𝐴 (𝑅1 “ On)))
3 rankidn 8932 . . . . 5 (𝐴 (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
42, 3syl6 35 . . . 4 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
54imp 395 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴)))
6 rankon 8905 . . . . 5 (rank‘𝐵) ∈ On
7 rankon 8905 . . . . 5 (rank‘𝐴) ∈ On
8 ontri1 5970 . . . . 5 (((rank‘𝐵) ∈ On ∧ (rank‘𝐴) ∈ On) → ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵)))
96, 7, 8mp2an 675 . . . 4 ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ (rank‘𝐵))
10 rankdmr1 8911 . . . . . 6 (rank‘𝐵) ∈ dom 𝑅1
11 rankdmr1 8911 . . . . . 6 (rank‘𝐴) ∈ dom 𝑅1
12 r1ord3g 8889 . . . . . 6 (((rank‘𝐵) ∈ dom 𝑅1 ∧ (rank‘𝐴) ∈ dom 𝑅1) → ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴))))
1310, 11, 12mp2an 675 . . . . 5 ((rank‘𝐵) ⊆ (rank‘𝐴) → (𝑅1‘(rank‘𝐵)) ⊆ (𝑅1‘(rank‘𝐴)))
14 r1rankidb 8914 . . . . . . 7 (𝐵 (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
1514sselda 3798 . . . . . 6 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐵)))
16 ssel 3792 . . . . . 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 234 . . 3 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (¬ (rank‘𝐴) ∈ (rank‘𝐵) → 𝐴 ∈ (𝑅1‘(rank‘𝐴))))
205, 19mt3d 142 . 2 ((𝐵 (𝑅1 “ On) ∧ 𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
2120ex 399 1 (𝐵 (𝑅1 “ On) → (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wcel 2156  wss 3769   cuni 4630  dom cdm 5311  cima 5314  Oncon0 5936  cfv 6101  𝑅1cr1 8872  rankcrnk 8873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7296  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-r1 8874  df-rank 8875
This theorem is referenced by:  wfelirr  8935  rankval3b  8936  rankel  8949  rankunb  8960  rankuni2b  8963  rankcf  9884
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