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Theorem oaf1o 6806
Description: Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
oaf1o  |-  ( A  e.  On  ->  (
x  e.  On  |->  ( A  +o  x ) ) : On -1-1-onto-> ( On  \  A
) )
Distinct variable group:    x, A

Proof of Theorem oaf1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 oacl 6779 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  e.  On )
2 oaword1 6795 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  On )  ->  A  C_  ( A  +o  x ) )
3 ontri1 4615 . . . . . 6  |-  ( ( A  e.  On  /\  ( A  +o  x
)  e.  On )  ->  ( A  C_  ( A  +o  x
)  <->  -.  ( A  +o  x )  e.  A
) )
41, 3syldan 457 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  C_  ( A  +o  x )  <->  -.  ( A  +o  x )  e.  A ) )
52, 4mpbid 202 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  -.  ( A  +o  x )  e.  A
)
61, 5eldifd 3331 . . 3  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  e.  ( On 
\  A ) )
76ralrimiva 2789 . 2  |-  ( A  e.  On  ->  A. x  e.  On  ( A  +o  x )  e.  ( On  \  A ) )
8 simpl 444 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  A  e.  On )
9 eldifi 3469 . . . . . 6  |-  ( y  e.  ( On  \  A )  ->  y  e.  On )
109adantl 453 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  -> 
y  e.  On )
11 eldifn 3470 . . . . . . 7  |-  ( y  e.  ( On  \  A )  ->  -.  y  e.  A )
1211adantl 453 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  -.  y  e.  A
)
13 ontri1 4615 . . . . . . 7  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  C_  y  <->  -.  y  e.  A ) )
1410, 13syldan 457 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  -> 
( A  C_  y  <->  -.  y  e.  A ) )
1512, 14mpbird 224 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  A  C_  y )
16 oawordeu 6798 . . . . 5  |-  ( ( ( A  e.  On  /\  y  e.  On )  /\  A  C_  y
)  ->  E! x  e.  On  ( A  +o  x )  =  y )
178, 10, 15, 16syl21anc 1183 . . . 4  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  E! x  e.  On  ( A  +o  x
)  =  y )
18 eqcom 2438 . . . . 5  |-  ( ( A  +o  x )  =  y  <->  y  =  ( A  +o  x
) )
1918reubii 2894 . . . 4  |-  ( E! x  e.  On  ( A  +o  x )  =  y  <->  E! x  e.  On  y  =  ( A  +o  x ) )
2017, 19sylib 189 . . 3  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  E! x  e.  On  y  =  ( A  +o  x ) )
2120ralrimiva 2789 . 2  |-  ( A  e.  On  ->  A. y  e.  ( On  \  A
) E! x  e.  On  y  =  ( A  +o  x ) )
22 eqid 2436 . . 3  |-  ( x  e.  On  |->  ( A  +o  x ) )  =  ( x  e.  On  |->  ( A  +o  x ) )
2322f1ompt 5891 . 2  |-  ( ( x  e.  On  |->  ( A  +o  x ) ) : On -1-1-onto-> ( On  \  A
)  <->  ( A. x  e.  On  ( A  +o  x )  e.  ( On  \  A )  /\  A. y  e.  ( On  \  A
) E! x  e.  On  y  =  ( A  +o  x ) ) )
247, 21, 23sylanbrc 646 1  |-  ( A  e.  On  ->  (
x  e.  On  |->  ( A  +o  x ) ) : On -1-1-onto-> ( On  \  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E!wreu 2707    \ cdif 3317    C_ wss 3320    e. cmpt 4266   Oncon0 4581   -1-1-onto->wf1o 5453  (class class class)co 6081    +o coa 6721
This theorem is referenced by:  oacomf1olem  6807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-oadd 6728
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