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Theorem r1ord3 7450
Description: Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)
Assertion
Ref Expression
r1ord3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( R1 `  A
)  C_  ( R1 `  B ) ) )

Proof of Theorem r1ord3
StepHypRef Expression
1 r1fnon 7435 . . . 4  |-  R1  Fn  On
2 fndm 5309 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
31, 2ax-mp 8 . . 3  |-  dom  R1  =  On
43eleq2i 2348 . 2  |-  ( A  e.  dom  R1  <->  A  e.  On )
53eleq2i 2348 . 2  |-  ( B  e.  dom  R1  <->  B  e.  On )
6 r1ord3g 7447 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  C_  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
74, 5, 6syl2anbr 466 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( R1 `  A
)  C_  ( R1 `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685    C_ wss 3153   Oncon0 4391    dom cdm 4688    Fn wfn 5216   ` cfv 5221   R1cr1 7430
This theorem is referenced by:  bndrank  7509  rankval4  7535  aomclem2  26563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-r1 7432
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