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Theorem oaf1o 7204
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 7177 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  e.  On )
2 oaword1 7193 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  On )  ->  A  C_  ( A  +o  x ) )
3 ontri1 4901 . . . . . 6  |-  ( ( A  e.  On  /\  ( A  +o  x
)  e.  On )  ->  ( A  C_  ( A  +o  x
)  <->  -.  ( A  +o  x )  e.  A
) )
41, 3syldan 468 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  C_  ( A  +o  x )  <->  -.  ( A  +o  x )  e.  A ) )
52, 4mpbid 210 . . . 4  |-  ( ( A  e.  On  /\  x  e.  On )  ->  -.  ( A  +o  x )  e.  A
)
61, 5eldifd 3472 . . 3  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  +o  x
)  e.  ( On 
\  A ) )
76ralrimiva 2868 . 2  |-  ( A  e.  On  ->  A. x  e.  On  ( A  +o  x )  e.  ( On  \  A ) )
8 simpl 455 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  A  e.  On )
9 eldifi 3612 . . . . . 6  |-  ( y  e.  ( On  \  A )  ->  y  e.  On )
109adantl 464 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  -> 
y  e.  On )
11 eldifn 3613 . . . . . . 7  |-  ( y  e.  ( On  \  A )  ->  -.  y  e.  A )
1211adantl 464 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  -.  y  e.  A
)
13 ontri1 4901 . . . . . . 7  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  C_  y  <->  -.  y  e.  A ) )
1410, 13syldan 468 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  -> 
( A  C_  y  <->  -.  y  e.  A ) )
1512, 14mpbird 232 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  A  C_  y )
16 oawordeu 7196 . . . . 5  |-  ( ( ( A  e.  On  /\  y  e.  On )  /\  A  C_  y
)  ->  E! x  e.  On  ( A  +o  x )  =  y )
178, 10, 15, 16syl21anc 1225 . . . 4  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  E! x  e.  On  ( A  +o  x
)  =  y )
18 eqcom 2463 . . . . 5  |-  ( ( A  +o  x )  =  y  <->  y  =  ( A  +o  x
) )
1918reubii 3041 . . . 4  |-  ( E! x  e.  On  ( A  +o  x )  =  y  <->  E! x  e.  On  y  =  ( A  +o  x ) )
2017, 19sylib 196 . . 3  |-  ( ( A  e.  On  /\  y  e.  ( On  \  A ) )  ->  E! x  e.  On  y  =  ( A  +o  x ) )
2120ralrimiva 2868 . 2  |-  ( A  e.  On  ->  A. y  e.  ( On  \  A
) E! x  e.  On  y  =  ( A  +o  x ) )
22 eqid 2454 . . 3  |-  ( x  e.  On  |->  ( A  +o  x ) )  =  ( x  e.  On  |->  ( A  +o  x ) )
2322f1ompt 6029 . 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 662 1  |-  ( A  e.  On  ->  (
x  e.  On  |->  ( A  +o  x ) ) : On -1-1-onto-> ( On  \  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E!wreu 2806    \ cdif 3458    C_ wss 3461    |-> cmpt 4497   Oncon0 4867   -1-1-onto->wf1o 5569  (class class class)co 6270    +o coa 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-oadd 7126
This theorem is referenced by:  oacomf1olem  7205
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