| Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > limexissupab | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| limexissupab | ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | limuni 6445 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = ∪ 𝐴) |
| 3 | limord 6444 | . . . 4 ⊢ (Lim 𝐴 → Ord 𝐴) | |
| 4 | ordsson 7803 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ (Lim 𝐴 → 𝐴 ⊆ On) |
| 6 | onsupuni 43241 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴) | |
| 7 | 5, 6 | sylan 580 | . 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴) |
| 8 | abid1 2878 | . . 3 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
| 9 | supeq1 9485 | . . 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 2783 | 1 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ⊆ wss 3951 ∪ cuni 4907 E cep 5583 Ord word 6383 Oncon0 6384 Lim wlim 6385 supcsup 9480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-iota 6514 df-riota 7388 df-sup 9482 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |