Theorem limexissupab 44284

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 6425	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2	1	adantr 486	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = ∪ 𝐴)
3		limord 6424	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
4		ordsson 7797	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	syl 18	. . 3 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
6		onsupuni 44230	. . 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 2897	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
9		supeq1 9437	. . 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 2802	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 2739   ⊆ wss 3899  ∪ cuni 4867   E cep 5550  Ord word 6361  Oncon0 6362  Lim wlim 6363  supcsup 9432

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 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6365  df-on 6366  df-lim 6367  df-iota 6494  df-riota 7377  df-sup 9434

This theorem is used by: (None)