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Theorem rankelb 9818
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 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ (𝐴 ∈ 𝐡 β†’ (rankβ€˜π΄) ∈ (rankβ€˜π΅)))

Proof of Theorem rankelb
StepHypRef Expression
1 r1elssi 9799 . . . . . 6 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝐡 βŠ† βˆͺ (𝑅1 β€œ On))
21sseld 3981 . . . . 5 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ (𝐴 ∈ 𝐡 β†’ 𝐴 ∈ βˆͺ (𝑅1 β€œ On)))
3 rankidn 9816 . . . . 5 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ Β¬ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄)))
42, 3syl6 35 . . . 4 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ (𝐴 ∈ 𝐡 β†’ Β¬ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄))))
54imp 407 . . 3 ((𝐡 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ 𝐡) β†’ Β¬ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄)))
6 rankon 9789 . . . . 5 (rankβ€˜π΅) ∈ On
7 rankon 9789 . . . . 5 (rankβ€˜π΄) ∈ On
8 ontri1 6398 . . . . 5 (((rankβ€˜π΅) ∈ On ∧ (rankβ€˜π΄) ∈ On) β†’ ((rankβ€˜π΅) βŠ† (rankβ€˜π΄) ↔ Β¬ (rankβ€˜π΄) ∈ (rankβ€˜π΅)))
96, 7, 8mp2an 690 . . . 4 ((rankβ€˜π΅) βŠ† (rankβ€˜π΄) ↔ Β¬ (rankβ€˜π΄) ∈ (rankβ€˜π΅))
10 rankdmr1 9795 . . . . . 6 (rankβ€˜π΅) ∈ dom 𝑅1
11 rankdmr1 9795 . . . . . 6 (rankβ€˜π΄) ∈ dom 𝑅1
12 r1ord3g 9773 . . . . . 6 (((rankβ€˜π΅) ∈ dom 𝑅1 ∧ (rankβ€˜π΄) ∈ dom 𝑅1) β†’ ((rankβ€˜π΅) βŠ† (rankβ€˜π΄) β†’ (𝑅1β€˜(rankβ€˜π΅)) βŠ† (𝑅1β€˜(rankβ€˜π΄))))
1310, 11, 12mp2an 690 . . . . 5 ((rankβ€˜π΅) βŠ† (rankβ€˜π΄) β†’ (𝑅1β€˜(rankβ€˜π΅)) βŠ† (𝑅1β€˜(rankβ€˜π΄)))
14 r1rankidb 9798 . . . . . 6 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝐡 βŠ† (𝑅1β€˜(rankβ€˜π΅)))
1514sselda 3982 . . . . 5 ((𝐡 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΅)))
16 ssel 3975 . . . . 5 ((𝑅1β€˜(rankβ€˜π΅)) βŠ† (𝑅1β€˜(rankβ€˜π΄)) β†’ (𝐴 ∈ (𝑅1β€˜(rankβ€˜π΅)) β†’ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄))))
1713, 15, 16syl2imc 41 . . . 4 ((𝐡 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ 𝐡) β†’ ((rankβ€˜π΅) βŠ† (rankβ€˜π΄) β†’ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄))))
189, 17biimtrrid 242 . . 3 ((𝐡 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ 𝐡) β†’ (Β¬ (rankβ€˜π΄) ∈ (rankβ€˜π΅) β†’ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄))))
195, 18mt3d 148 . 2 ((𝐡 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ 𝐡) β†’ (rankβ€˜π΄) ∈ (rankβ€˜π΅))
2019ex 413 1 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ (𝐴 ∈ 𝐡 β†’ (rankβ€˜π΄) ∈ (rankβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   βŠ† wss 3948  βˆͺ cuni 4908  dom cdm 5676   β€œ cima 5679  Oncon0 6364  β€˜cfv 6543  π‘…1cr1 9756  rankcrnk 9757
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-r1 9758  df-rank 9759
This theorem is referenced by:  wfelirr  9819  rankval3b  9820  rankel  9833  rankunb  9844  rankuni2b  9847  rankcf  10771
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