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Theorem r1ord3g 7661
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
r1ord3g  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  C_  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )

Proof of Theorem r1ord3g
StepHypRef Expression
1 r1funlim 7648 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 449 . . . . 5  |-  Lim  dom  R1
3 limord 4600 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 4729 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
65sseli 3304 . . 3  |-  ( A  e.  dom  R1  ->  A  e.  On )
75sseli 3304 . . 3  |-  ( B  e.  dom  R1  ->  B  e.  On )
8 onsseleq 4582 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
96, 7, 8syl2an 464 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  C_  B 
<->  ( A  e.  B  \/  A  =  B
) ) )
10 r1ordg 7660 . . . . 5  |-  ( B  e.  dom  R1  ->  ( A  e.  B  -> 
( R1 `  A
)  e.  ( R1
`  B ) ) )
1110adantl 453 . . . 4  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
12 r1tr 7658 . . . . 5  |-  Tr  ( R1 `  B )
13 trss 4271 . . . . 5  |-  ( Tr  ( R1 `  B
)  ->  ( ( R1 `  A )  e.  ( R1 `  B
)  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
1412, 13ax-mp 8 . . . 4  |-  ( ( R1 `  A )  e.  ( R1 `  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) )
1511, 14syl6 31 . . 3  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  e.  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
16 fveq2 5687 . . . . 5  |-  ( A  =  B  ->  ( R1 `  A )  =  ( R1 `  B
) )
17 eqimss 3360 . . . . 5  |-  ( ( R1 `  A )  =  ( R1 `  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) )
1816, 17syl 16 . . . 4  |-  ( A  =  B  ->  ( R1 `  A )  C_  ( R1 `  B ) )
1918a1i 11 . . 3  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  =  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
2015, 19jaod 370 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( ( A  e.  B  \/  A  =  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
219, 20sylbid 207 1  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  C_  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   Tr wtr 4262   Ord word 4540   Oncon0 4541   Lim wlim 4542   dom cdm 4837   Fun wfun 5407   ` cfv 5413   R1cr1 7644
This theorem is referenced by:  r1ord3  7664  r1val1  7668  rankr1ag  7684  unwf  7692  rankelb  7706  rankonidlem  7710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646
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