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Theorem oaf1o 6561
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 6534 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  e.  On )
2 oaword1 6550 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  On )  ->  A  C_  ( A  +o  x ) )
3 ontri1 4426 . . . . . 6  |-  ( ( A  e.  On  /\  ( A  +o  x
)  e.  On )  ->  ( A  C_  ( A  +o  x
)  <->  -.  ( A  +o  x )  e.  A
) )
41, 3syldan 456 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  C_  ( A  +o  x )  <->  -.  ( A  +o  x )  e.  A ) )
52, 4mpbid 201 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  -.  ( A  +o  x )  e.  A
)
6 eldif 3162 . . . 4  |-  ( ( A  +o  x )  e.  ( On  \  A )  <->  ( ( A  +o  x )  e.  On  /\  -.  ( A  +o  x )  e.  A ) )
71, 5, 6sylanbrc 645 . . 3  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  e.  ( On 
\  A ) )
87ralrimiva 2626 . 2  |-  ( A  e.  On  ->  A. x  e.  On  ( A  +o  x )  e.  ( On  \  A ) )
9 simpl 443 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  A  e.  On )
10 eldifi 3298 . . . . . 6  |-  ( y  e.  ( On  \  A )  ->  y  e.  On )
1110adantl 452 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  -> 
y  e.  On )
12 eldifn 3299 . . . . . . 7  |-  ( y  e.  ( On  \  A )  ->  -.  y  e.  A )
1312adantl 452 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  -.  y  e.  A
)
14 ontri1 4426 . . . . . . 7  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  C_  y  <->  -.  y  e.  A ) )
1511, 14syldan 456 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  -> 
( A  C_  y  <->  -.  y  e.  A ) )
1613, 15mpbird 223 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  A  C_  y )
17 oawordeu 6553 . . . . 5  |-  ( ( ( A  e.  On  /\  y  e.  On )  /\  A  C_  y
)  ->  E! x  e.  On  ( A  +o  x )  =  y )
189, 11, 16, 17syl21anc 1181 . . . 4  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  E! x  e.  On  ( A  +o  x
)  =  y )
19 eqcom 2285 . . . . 5  |-  ( ( A  +o  x )  =  y  <->  y  =  ( A  +o  x
) )
2019reubii 2726 . . . 4  |-  ( E! x  e.  On  ( A  +o  x )  =  y  <->  E! x  e.  On  y  =  ( A  +o  x ) )
2118, 20sylib 188 . . 3  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  E! x  e.  On  y  =  ( A  +o  x ) )
2221ralrimiva 2626 . 2  |-  ( A  e.  On  ->  A. y  e.  ( On  \  A
) E! x  e.  On  y  =  ( A  +o  x ) )
23 eqid 2283 . . 3  |-  ( x  e.  On  |->  ( A  +o  x ) )  =  ( x  e.  On  |->  ( A  +o  x ) )
2423f1ompt 5682 . 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 ) ) )
258, 22, 24sylanbrc 645 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E!wreu 2545    \ cdif 3149    C_ wss 3152    e. cmpt 4077   Oncon0 4392   -1-1-onto->wf1o 5254  (class class class)co 5858    +o coa 6476
This theorem is referenced by:  oacomf1olem  6562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483
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