Theorem limexissupab 44132

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 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem limexissupab

Step	Hyp	Ref	Expression
1		limuni 6424	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2	1	adantr 486	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = ∪ 𝐴)
3		limord 6423	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
4		ordsson 7786	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	syl 18	. . 3 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
6		onsupuni 44078	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)
7	5, 6	sylan 592	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)
8		abid1 2898	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
9		supeq1 9419	. . 3 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} → sup(𝐴, On, E ) = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))
10	8, 9	mp1i 14	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))
11	2, 7, 10	3eqtr2d 2803	1 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740   ⊆ wss 3902  ∪ cuni 4870   E cep 5558  Ord word 6360  Oncon0 6361  Lim wlim 6362  supcsup 9414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365  df-lim 6366  df-iota 6493  df-riota 7374  df-sup 9416

This theorem is used by: (None)