Theorem limexissupab 43640

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 6387	. . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
2	1	adantr 480	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = ∪ 𝐴)
3		limord 6386	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
4		ordsson 7738	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5	3, 4	syl 17	. . 3 ⊢ (Lim 𝐴 → 𝐴 ⊆ On)
6		onsupuni 43586	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)
7	5, 6	sylan 581	. 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)
8		abid1 2873	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴}
9		supeq1 9360	. . 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 2778	1 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715   ⊆ wss 3903  ∪ cuni 4865   E cep 5531  Ord word 6324  Oncon0 6325  Lim wlim 6326  supcsup 9355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-lim 6330  df-iota 6456  df-riota 7325  df-sup 9357

This theorem is referenced by: (None)