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Theorem rankelb 9768
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 9749 . . . . . 6 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝐡 βŠ† βˆͺ (𝑅1 β€œ On))
21sseld 3947 . . . . 5 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ (𝐴 ∈ 𝐡 β†’ 𝐴 ∈ βˆͺ (𝑅1 β€œ On)))
3 rankidn 9766 . . . . 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 408 . . 3 ((𝐡 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ 𝐡) β†’ Β¬ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΄)))
6 rankon 9739 . . . . 5 (rankβ€˜π΅) ∈ On
7 rankon 9739 . . . . 5 (rankβ€˜π΄) ∈ On
8 ontri1 6355 . . . . 5 (((rankβ€˜π΅) ∈ On ∧ (rankβ€˜π΄) ∈ On) β†’ ((rankβ€˜π΅) βŠ† (rankβ€˜π΄) ↔ Β¬ (rankβ€˜π΄) ∈ (rankβ€˜π΅)))
96, 7, 8mp2an 691 . . . 4 ((rankβ€˜π΅) βŠ† (rankβ€˜π΄) ↔ Β¬ (rankβ€˜π΄) ∈ (rankβ€˜π΅))
10 rankdmr1 9745 . . . . . 6 (rankβ€˜π΅) ∈ dom 𝑅1
11 rankdmr1 9745 . . . . . 6 (rankβ€˜π΄) ∈ dom 𝑅1
12 r1ord3g 9723 . . . . . 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 691 . . . . 5 ((rankβ€˜π΅) βŠ† (rankβ€˜π΄) β†’ (𝑅1β€˜(rankβ€˜π΅)) βŠ† (𝑅1β€˜(rankβ€˜π΄)))
14 r1rankidb 9748 . . . . . 6 (𝐡 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝐡 βŠ† (𝑅1β€˜(rankβ€˜π΅)))
1514sselda 3948 . . . . 5 ((𝐡 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 ∈ (𝑅1β€˜(rankβ€˜π΅)))
16 ssel 3941 . . . . 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 414 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 397   ∈ wcel 2107   βŠ† wss 3914  βˆͺ cuni 4869  dom cdm 5637   β€œ cima 5640  Oncon0 6321  β€˜cfv 6500  π‘…1cr1 9706  rankcrnk 9707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-r1 9708  df-rank 9709
This theorem is referenced by:  wfelirr  9769  rankval3b  9770  rankel  9783  rankunb  9794  rankuni2b  9797  rankcf  10721
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