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Theorem ssrankr1 9825
Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets 𝑅1. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypothesis
Ref Expression
rankid.1 𝐴 ∈ V
Assertion
Ref Expression
ssrankr1 (𝐡 ∈ On β†’ (𝐡 βŠ† (rankβ€˜π΄) ↔ Β¬ 𝐴 ∈ (𝑅1β€˜π΅)))

Proof of Theorem ssrankr1
StepHypRef Expression
1 rankid.1 . . . 4 𝐴 ∈ V
2 unir1 9803 . . . 4 βˆͺ (𝑅1 β€œ On) = V
31, 2eleqtrri 2824 . . 3 𝐴 ∈ βˆͺ (𝑅1 β€œ On)
4 r1fnon 9757 . . . . . 6 𝑅1 Fn On
5 fndm 6642 . . . . . 6 (𝑅1 Fn On β†’ dom 𝑅1 = On)
64, 5ax-mp 5 . . . . 5 dom 𝑅1 = On
76eleq2i 2817 . . . 4 (𝐡 ∈ dom 𝑅1 ↔ 𝐡 ∈ On)
87biimpri 227 . . 3 (𝐡 ∈ On β†’ 𝐡 ∈ dom 𝑅1)
9 rankr1clem 9810 . . 3 ((𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐡 ∈ dom 𝑅1) β†’ (Β¬ 𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝐡 βŠ† (rankβ€˜π΄)))
103, 8, 9sylancr 586 . 2 (𝐡 ∈ On β†’ (Β¬ 𝐴 ∈ (𝑅1β€˜π΅) ↔ 𝐡 βŠ† (rankβ€˜π΄)))
1110bicomd 222 1 (𝐡 ∈ On β†’ (𝐡 βŠ† (rankβ€˜π΄) ↔ Β¬ 𝐴 ∈ (𝑅1β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3466   βŠ† wss 3940  βˆͺ cuni 4899  dom cdm 5666   β€œ cima 5669  Oncon0 6354   Fn wfn 6528  β€˜cfv 6533  π‘…1cr1 9752  rankcrnk 9753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-reg 9582  ax-inf2 9631
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-r1 9754  df-rank 9755
This theorem is referenced by:  rankr1a  9826
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