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Theorem unizlim 5882
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 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))

Proof of Theorem unizlim
StepHypRef Expression
1 df-ne 2824 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 df-lim 5766 . . . . . . . . 9 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
32biimpri 218 . . . . . . . 8 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) → Lim 𝐴)
433exp 1283 . . . . . . 7 (Ord 𝐴 → (𝐴 ≠ ∅ → (𝐴 = 𝐴 → Lim 𝐴)))
51, 4syl5bir 233 . . . . . 6 (Ord 𝐴 → (¬ 𝐴 = ∅ → (𝐴 = 𝐴 → Lim 𝐴)))
65com23 86 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 → (¬ 𝐴 = ∅ → Lim 𝐴)))
76imp 444 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴))
87orrd 392 . . 3 ((Ord 𝐴𝐴 = 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴))
98ex 449 . 2 (Ord 𝐴 → (𝐴 = 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴)))
10 uni0 4497 . . . . 5 ∅ = ∅
1110eqcomi 2660 . . . 4 ∅ =
12 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
13 unieq 4476 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1411, 12, 133eqtr4a 2711 . . 3 (𝐴 = ∅ → 𝐴 = 𝐴)
15 limuni 5823 . . 3 (Lim 𝐴𝐴 = 𝐴)
1614, 15jaoi 393 . 2 ((𝐴 = ∅ ∨ Lim 𝐴) → 𝐴 = 𝐴)
179, 16impbid1 215 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wne 2823  c0 3948   cuni 4468  Ord word 5760  Lim wlim 5762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469  df-lim 5766
This theorem is referenced by:  ordzsl  7087  oeeulem  7726  cantnfp1lem2  8614  cantnflem1  8624  cnfcom2lem  8636  ordcmp  32571
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