Theorem limexissupab 43728

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 6372	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2	1	adantr 481	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = ∪ 𝐴)
3		limord 6371	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
4		ordsson 7726	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	syl 17	. . 3 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
6		onsupuni 43674	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)
7	5, 6	sylan 586	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)
8		abid1 2875	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
9		supeq1 9348	. . 3 ⊢ (𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} → sup(𝐴, On, E ) = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))
10	8, 9	mp1i 13	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))
11	2, 7, 10	3eqtr2d 2780	1 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717   ⊆ wss 3883  ∪ cuni 4838   E cep 5517  Ord word 6309  Oncon0 6310  Lim wlim 6311  supcsup 9343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-lim 6315  df-iota 6441  df-riota 7313  df-sup 9345

This theorem is referenced by: (None)