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Theorem limexissupab 43273
Description: An ordinal which is a limit ordinal is equal to the supremum of the class of all its elements. Lemma 2.13 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
limexissupab ((Lim 𝐴𝐴𝑉) → 𝐴 = sup({𝑥𝑥𝐴}, On, E ))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem limexissupab
StepHypRef Expression
1 limuni 6447 . . 3 (Lim 𝐴𝐴 = 𝐴)
21adantr 480 . 2 ((Lim 𝐴𝐴𝑉) → 𝐴 = 𝐴)
3 limord 6446 . . . 4 (Lim 𝐴 → Ord 𝐴)
4 ordsson 7802 . . . 4 (Ord 𝐴𝐴 ⊆ On)
53, 4syl 17 . . 3 (Lim 𝐴𝐴 ⊆ On)
6 onsupuni 43218 . . 3 ((𝐴 ⊆ On ∧ 𝐴𝑉) → sup(𝐴, On, E ) = 𝐴)
75, 6sylan 580 . 2 ((Lim 𝐴𝐴𝑉) → sup(𝐴, On, E ) = 𝐴)
8 abid1 2876 . . 3 𝐴 = {𝑥𝑥𝐴}
9 supeq1 9483 . . 3 (𝐴 = {𝑥𝑥𝐴} → sup(𝐴, On, E ) = sup({𝑥𝑥𝐴}, On, E ))
108, 9mp1i 13 . 2 ((Lim 𝐴𝐴𝑉) → sup(𝐴, On, E ) = sup({𝑥𝑥𝐴}, On, E ))
112, 7, 103eqtr2d 2781 1 ((Lim 𝐴𝐴𝑉) → 𝐴 = sup({𝑥𝑥𝐴}, On, E ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wss 3963   cuni 4912   E cep 5588  Ord word 6385  Oncon0 6386  Lim wlim 6387  supcsup 9478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-iota 6516  df-riota 7388  df-sup 9480
This theorem is referenced by: (None)
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