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Theorem nnawordex 6390
Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnawordex  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  C_  B  <->  E. x  e.  om  ( A  +o  x )  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem nnawordex
StepHypRef Expression
1 nntri3or 6355 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )
213adant3 984 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A )
)
3 nnaordex 6389 . . . . . . 7  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  <->  E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
4 simpr 109 . . . . . . . 8  |-  ( (
(/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  ( A  +o  x )  =  B )
54reximi 2504 . . . . . . 7  |-  ( E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B )  ->  E. x  e.  om  ( A  +o  x )  =  B )
63, 5syl6bi 162 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  E. x  e.  om  ( A  +o  x
)  =  B ) )
763adant3 984 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  ( A  e.  B  ->  E. x  e.  om  ( A  +o  x )  =  B ) )
8 nna0 6336 . . . . . . . 8  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
983ad2ant1 985 . . . . . . 7  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  ( A  +o  (/) )  =  A )
10 eqeq2 2125 . . . . . . 7  |-  ( A  =  B  ->  (
( A  +o  (/) )  =  A  <->  ( A  +o  (/) )  =  B ) )
119, 10syl5ibcom 154 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  ( A  =  B  ->  ( A  +o  (/) )  =  B ) )
12 peano1 4476 . . . . . . 7  |-  (/)  e.  om
13 oveq2 5748 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( A  +o  x )  =  ( A  +o  (/) ) )
1413eqeq1d 2124 . . . . . . . 8  |-  ( x  =  (/)  ->  ( ( A  +o  x )  =  B  <->  ( A  +o  (/) )  =  B ) )
1514rspcev 2761 . . . . . . 7  |-  ( (
(/)  e.  om  /\  ( A  +o  (/) )  =  B )  ->  E. x  e.  om  ( A  +o  x )  =  B )
1612, 15mpan 418 . . . . . 6  |-  ( ( A  +o  (/) )  =  B  ->  E. x  e.  om  ( A  +o  x )  =  B )
1711, 16syl6 33 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  ( A  =  B  ->  E. x  e.  om  ( A  +o  x )  =  B ) )
18 nntri1 6358 . . . . . . 7  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
1918biimp3a 1306 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  -.  B  e.  A )
2019pm2.21d 591 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  ( B  e.  A  ->  E. x  e.  om  ( A  +o  x )  =  B ) )
217, 17, 203jaod 1265 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  (
( A  e.  B  \/  A  =  B  \/  B  e.  A
)  ->  E. x  e.  om  ( A  +o  x )  =  B ) )
222, 21mpd 13 . . 3  |-  ( ( A  e.  om  /\  B  e.  om  /\  A  C_  B )  ->  E. x  e.  om  ( A  +o  x )  =  B )
23223expia 1166 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  C_  B  ->  E. x  e.  om  ( A  +o  x
)  =  B ) )
24 nnaword1 6375 . . . . 5  |-  ( ( A  e.  om  /\  x  e.  om )  ->  A  C_  ( A  +o  x ) )
25 sseq2 3089 . . . . 5  |-  ( ( A  +o  x )  =  B  ->  ( A  C_  ( A  +o  x )  <->  A  C_  B
) )
2624, 25syl5ibcom 154 . . . 4  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  A  C_  B
) )
2726rexlimdva 2524 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  om  ( A  +o  x
)  =  B  ->  A  C_  B ) )
2827adantr 272 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. x  e. 
om  ( A  +o  x )  =  B  ->  A  C_  B
) )
2923, 28impbid 128 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  C_  B  <->  E. x  e.  om  ( A  +o  x )  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 944    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392    C_ wss 3039   (/)c0 3331   omcom 4472  (class class class)co 5740    +o coa 6276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283
This theorem is referenced by:  prarloclemn  7271
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