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Theorem r1ord3g 7419
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 7406 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 450 . . . . 5  |-  Lim  dom  R1
3 limord 4423 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 4553 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 11 . . . 4  |-  dom  R1  C_  On
65sseli 3151 . . 3  |-  ( A  e.  dom  R1  ->  A  e.  On )
75sseli 3151 . . 3  |-  ( B  e.  dom  R1  ->  B  e.  On )
8 onsseleq 4405 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
96, 7, 8syl2an 465 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  C_  B 
<->  ( A  e.  B  \/  A  =  B
) ) )
10 r1ordg 7418 . . . . 5  |-  ( B  e.  dom  R1  ->  ( A  e.  B  -> 
( R1 `  A
)  e.  ( R1
`  B ) ) )
1110adantl 454 . . . 4  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
12 r1tr 7416 . . . . 5  |-  Tr  ( R1 `  B )
13 trss 4096 . . . . 5  |-  ( Tr  ( R1 `  B
)  ->  ( ( R1 `  A )  e.  ( R1 `  B
)  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
1412, 13ax-mp 10 . . . 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 5458 . . . . 5  |-  ( A  =  B  ->  ( R1 `  A )  =  ( R1 `  B
) )
17 eqimss 3205 . . . . 5  |-  ( ( R1 `  A )  =  ( R1 `  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) )
1816, 17syl 17 . . . 4  |-  ( A  =  B  ->  ( R1 `  A )  C_  ( R1 `  B ) )
1918a1i 12 . . 3  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( A  =  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
2015, 19jaod 371 . 2  |-  ( ( A  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( ( A  e.  B  \/  A  =  B )  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
219, 20sylbid 208 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 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3127   Tr wtr 4087   Ord word 4363   Oncon0 4364   Lim wlim 4365   dom cdm 4661   Fun wfun 4667   ` cfv 4673   R1cr1 7402
This theorem is referenced by:  r1ord3  7422  r1val1  7426  rankr1ag  7442  unwf  7450  rankelb  7464  rankonidlem  7468
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391  df-r1 7404
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