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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 6447 | . . 3 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴) | |
2 | 1 | adantr 480 | . 2 ⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = ∪ 𝐴) |
3 | limord 6446 | . . . 4 ⊢ (Lim 𝐴 → Ord 𝐴) | |
4 | ordsson 7802 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (Lim 𝐴 → 𝐴 ⊆ On) |
6 | onsupuni 43218 | . . 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 2876 | . . 3 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
9 | supeq1 9483 | . . 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 2781 | 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 1537 ∈ wcel 2106 {cab 2712 ⊆ wss 3963 ∪ cuni 4912 E cep 5588 Ord word 6385 Oncon0 6386 Lim wlim 6387 supcsup 9478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-lim 6391 df-iota 6516 df-riota 7388 df-sup 9480 |
This theorem is referenced by: (None) |
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