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Theorem r1ord3g 8258
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 8245 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 463 . . . . 5  |-  Lim  dom  R1
3 limord 5501 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 6630 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
65sseli 3460 . . 3  |-  ( A  e.  dom  R1  ->  A  e.  On )
75sseli 3460 . . 3  |-  ( B  e.  dom  R1  ->  B  e.  On )
8 onsseleq 5483 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
96, 7, 8syl2an 479 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  C_  B 
<->  ( A  e.  B  \/  A  =  B
) ) )
10 r1ordg 8257 . . . . 5  |-  ( B  e.  dom  R1  ->  ( A  e.  B  -> 
( R1 `  A
)  e.  ( R1
`  B ) ) )
1110adantl 467 . . . 4  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
12 r1tr 8255 . . . . 5  |-  Tr  ( R1 `  B )
13 trss 4527 . . . . 5  |-  ( Tr  ( R1 `  B
)  ->  ( ( R1 `  A )  e.  ( R1 `  B
)  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
1412, 13ax-mp 5 . . . 4  |-  ( ( R1 `  A )  e.  ( R1 `  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) )
1511, 14syl6 34 . . 3  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  e.  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
16 fveq2 5881 . . . . 5  |-  ( A  =  B  ->  ( R1 `  A )  =  ( R1 `  B
) )
17 eqimss 3516 . . . . 5  |-  ( ( R1 `  A )  =  ( R1 `  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) )
1816, 17syl 17 . . . 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 381 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( ( A  e.  B  \/  A  =  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
219, 20sylbid 218 1  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  C_  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436   Tr wtr 4518   dom cdm 4853   Ord word 5441   Oncon0 5442   Lim wlim 5443   Fun wfun 5595   ` cfv 5601   R1cr1 8241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-r1 8243
This theorem is referenced by:  r1ord3  8261  r1val1  8265  rankr1ag  8281  unwf  8289  rankelb  8303  rankonidlem  8307
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