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Theorem r1ord3g 8214
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 8201 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 462 . . . . 5  |-  Lim  dom  R1
3 limord 4946 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 6624 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
65sseli 3495 . . 3  |-  ( A  e.  dom  R1  ->  A  e.  On )
75sseli 3495 . . 3  |-  ( B  e.  dom  R1  ->  B  e.  On )
8 onsseleq 4928 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
96, 7, 8syl2an 477 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  C_  B 
<->  ( A  e.  B  \/  A  =  B
) ) )
10 r1ordg 8213 . . . . 5  |-  ( B  e.  dom  R1  ->  ( A  e.  B  -> 
( R1 `  A
)  e.  ( R1
`  B ) ) )
1110adantl 466 . . . 4  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
12 r1tr 8211 . . . . 5  |-  Tr  ( R1 `  B )
13 trss 4559 . . . . 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 33 . . 3  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  e.  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
16 fveq2 5872 . . . . 5  |-  ( A  =  B  ->  ( R1 `  A )  =  ( R1 `  B
) )
17 eqimss 3551 . . . . 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 380 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( ( A  e.  B  \/  A  =  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
219, 20sylbid 215 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 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    C_ wss 3471   Tr wtr 4550   Ord word 4886   Oncon0 4887   Lim wlim 4888   dom cdm 5008   Fun wfun 5588   ` cfv 5594   R1cr1 8197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199
This theorem is referenced by:  r1ord3  8217  r1val1  8221  rankr1ag  8237  unwf  8245  rankelb  8259  rankonidlem  8263
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