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Theorem unizlim 5803
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 2791 . . . . . . 7 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 df-lim 5687 . . . . . . . . 9 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
32biimpri 218 . . . . . . . 8 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) → Lim 𝐴)
433exp 1261 . . . . . . 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 445 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → (¬ 𝐴 = ∅ → Lim 𝐴))
87orrd 393 . . 3 ((Ord 𝐴𝐴 = 𝐴) → (𝐴 = ∅ ∨ Lim 𝐴))
98ex 450 . 2 (Ord 𝐴 → (𝐴 = 𝐴 → (𝐴 = ∅ ∨ Lim 𝐴)))
10 uni0 4431 . . . . 5 ∅ = ∅
1110eqcomi 2630 . . . 4 ∅ =
12 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
13 unieq 4410 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
1411, 12, 133eqtr4a 2681 . . 3 (𝐴 = ∅ → 𝐴 = 𝐴)
15 limuni 5744 . . 3 (Lim 𝐴𝐴 = 𝐴)
1614, 15jaoi 394 . 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 383  wa 384  w3a 1036   = wceq 1480  wne 2790  c0 3891   cuni 4402  Ord word 5681  Lim wlim 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-uni 4403  df-lim 5687
This theorem is referenced by:  ordzsl  6992  oeeulem  7626  cantnfp1lem2  8520  cantnflem1  8530  cnfcom2lem  8542  ordcmp  32085
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