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Theorem unizlim 4701
 Description: An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
Assertion
Ref Expression
unizlim

Proof of Theorem unizlim
StepHypRef Expression
1 df-ne 2603 . . . . . . 7
2 df-lim 4589 . . . . . . . . 9
32biimpri 199 . . . . . . . 8
433exp 1153 . . . . . . 7
51, 4syl5bir 211 . . . . . 6
65com23 75 . . . . 5
76imp 420 . . . 4
87orrd 369 . . 3
98ex 425 . 2
10 uni0 4044 . . . . 5
1110eqcomi 2442 . . . 4
12 id 21 . . . 4
13 unieq 4026 . . . 4
1411, 12, 133eqtr4a 2496 . . 3
15 limuni 4644 . . 3
1614, 15jaoi 370 . 2
179, 16impbid1 196 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wo 359   wa 360   w3a 937   wceq 1653   wne 2601  c0 3630  cuni 4017   word 4583   wlim 4585 This theorem is referenced by:  ordzsl  4828  oeeulem  6847  cantnfp1lem2  7638  cantnflem1  7648  cnfcom2lem  7661  ordcmp  26202 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-uni 4018  df-lim 4589
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