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Theorem onun2i 4697
 Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
Hypotheses
Ref Expression
on.1
on.2
Assertion
Ref Expression
onun2i

Proof of Theorem onun2i
StepHypRef Expression
1 on.2 . . . 4
21onordi 4686 . . 3
3 on.1 . . . 4
43onordi 4686 . . 3
5 ordtri2or 4677 . . 3
62, 4, 5mp2an 654 . 2
73oneluni 4694 . . . 4
87, 3syl6eqel 2524 . . 3
9 ssequn1 3517 . . . 4
10 eleq1 2496 . . . . 5
111, 10mpbiri 225 . . . 4
129, 11sylbi 188 . . 3
138, 12jaoi 369 . 2
146, 13ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wo 358   wceq 1652   wcel 1725   cun 3318   wss 3320   word 4580  con0 4581 This theorem is referenced by:  rankunb  7776  rankelun  7798  rankelpr  7799  inar1  8650 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 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-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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