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Theorem celsor 25214
 Description: If all the elements of a set are ordinal numbers and are parts of the set then is an ordinal number. (Contributed by FL, 20-Apr-2011.)
Assertion
Ref Expression
celsor
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem celsor
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.26 2688 . . . 4
2 dftr3 4133 . . . . . . 7
32biimpri 197 . . . . . 6
4 eleq1 2356 . . . . . . . . 9
54cbvralv 2777 . . . . . . . 8
6 raaanv 3575 . . . . . . . . . 10
7 eloni 4418 . . . . . . . . . . . . 13
8 eloni 4418 . . . . . . . . . . . . 13
9 ordtri3or 4440 . . . . . . . . . . . . 13
107, 8, 9syl2an 463 . . . . . . . . . . . 12
1110ralimi 2631 . . . . . . . . . . 11
1211ralimi 2631 . . . . . . . . . 10
136, 12sylbir 204 . . . . . . . . 9
1413expcom 424 . . . . . . . 8
155, 14sylbi 187 . . . . . . 7
1615pm2.43i 43 . . . . . 6
173, 16anim12i 549 . . . . 5
1817ancoms 439 . . . 4
191, 18sylbi 187 . . 3
2019adantl 452 . 2
21 elong 4416 . . . 4
2221adantr 451 . . 3
23 dford2 7337 . . 3
2422, 23syl6bb 252 . 2
2520, 24mpbird 223 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3o 933   wcel 1696  wral 2556   wss 3165   wtr 4129   word 4407  con0 4408 This theorem is referenced by:  inttaror  26003 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528  ax-reg 7322 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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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