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Mirrors > Home > MPE Home > Th. List > ssorduni | Structured version Visualization version GIF version |
Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
ssorduni | ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni2 4376 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
2 | ssel 3562 | . . . . . . . . 9 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On)) | |
3 | onelss 5683 | . . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)) | |
4 | 2, 3 | syl6 34 | . . . . . . . 8 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))) |
5 | anc2r 577 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)) → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴)))) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → (𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴)))) |
7 | ssuni 4395 | . . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ⊆ ∪ 𝐴) | |
8 | 6, 7 | syl8 74 | . . . . . 6 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴))) |
9 | 8 | rexlimdv 3012 | . . . . 5 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝐴)) |
10 | 1, 9 | syl5bi 231 | . . . 4 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ⊆ ∪ 𝐴)) |
11 | 10 | ralrimiv 2948 | . . 3 ⊢ (𝐴 ⊆ On → ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴) |
12 | dftr3 4684 | . . 3 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑥 ∈ ∪ 𝐴𝑥 ⊆ ∪ 𝐴) | |
13 | 11, 12 | sylibr 223 | . 2 ⊢ (𝐴 ⊆ On → Tr ∪ 𝐴) |
14 | onelon 5665 | . . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
15 | 14 | ex 449 | . . . . . 6 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ∈ On)) |
16 | 2, 15 | syl6 34 | . . . . 5 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ∈ On))) |
17 | 16 | rexlimdv 3012 | . . . 4 ⊢ (𝐴 ⊆ On → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → 𝑥 ∈ On)) |
18 | 1, 17 | syl5bi 231 | . . 3 ⊢ (𝐴 ⊆ On → (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ On)) |
19 | 18 | ssrdv 3574 | . 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ⊆ On) |
20 | ordon 6874 | . . 3 ⊢ Ord On | |
21 | trssord 5657 | . . . 4 ⊢ ((Tr ∪ 𝐴 ∧ ∪ 𝐴 ⊆ On ∧ Ord On) → Ord ∪ 𝐴) | |
22 | 21 | 3exp 1256 | . . 3 ⊢ (Tr ∪ 𝐴 → (∪ 𝐴 ⊆ On → (Ord On → Ord ∪ 𝐴))) |
23 | 20, 22 | mpii 45 | . 2 ⊢ (Tr ∪ 𝐴 → (∪ 𝐴 ⊆ On → Ord ∪ 𝐴)) |
24 | 13, 19, 23 | sylc 63 | 1 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∪ cuni 4372 Tr wtr 4680 Ord word 5639 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: ssonuni 6878 ssonprc 6884 orduni 6886 onsucuni 6920 limuni3 6944 onfununi 7325 tfrlem8 7367 onssnum 8746 unialeph 8807 cfslbn 8972 hsmexlem1 9131 inaprc 9537 nobndlem1 31091 nobndlem2 31092 |
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