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Mirrors > Home > MPE Home > Th. List > nnunifi | Structured version Visualization version GIF version |
Description: The union (supremum) of a finite set of finite ordinals is a finite ordinal. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
Ref | Expression |
---|---|
nnunifi | ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4923 | . . . 4 ⊢ (𝑆 = ∅ → ∪ 𝑆 = ∪ ∅) | |
2 | uni0 4942 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | peano1 7900 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | eqeltri 2825 | . . . 4 ⊢ ∪ ∅ ∈ ω |
5 | 1, 4 | eqeltrdi 2837 | . . 3 ⊢ (𝑆 = ∅ → ∪ 𝑆 ∈ ω) |
6 | 5 | adantl 480 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 = ∅) → ∪ 𝑆 ∈ ω) |
7 | simpll 765 | . . 3 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ω) | |
8 | omsson 7880 | . . . . 5 ⊢ ω ⊆ On | |
9 | 7, 8 | sstrdi 3994 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ On) |
10 | simplr 767 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ Fin) | |
11 | simpr 483 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅) | |
12 | ordunifi 9324 | . . . 4 ⊢ ((𝑆 ⊆ On ∧ 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ 𝑆) | |
13 | 9, 10, 11, 12 | syl3anc 1368 | . . 3 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ 𝑆) |
14 | 7, 13 | sseldd 3983 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ ω) |
15 | 6, 14 | pm2.61dane 3026 | 1 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ⊆ wss 3949 ∅c0 4326 ∪ cuni 4912 Oncon0 6374 ωcom 7876 Fincfn 8970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-en 8971 df-fin 8974 |
This theorem is referenced by: ackbij1lem16 10266 isf32lem5 10388 finxpreclem4 36906 |
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