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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 4681 | . . . 4 ⊢ (𝑆 = ∅ → ∪ 𝑆 = ∪ ∅) | |
2 | uni0 4702 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | peano1 7365 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | eqeltri 2855 | . . . 4 ⊢ ∪ ∅ ∈ ω |
5 | 1, 4 | syl6eqel 2867 | . . 3 ⊢ (𝑆 = ∅ → ∪ 𝑆 ∈ ω) |
6 | 5 | adantl 475 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 = ∅) → ∪ 𝑆 ∈ ω) |
7 | simpll 757 | . . 3 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ω) | |
8 | omsson 7349 | . . . . 5 ⊢ ω ⊆ On | |
9 | 7, 8 | syl6ss 3833 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ On) |
10 | simplr 759 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ Fin) | |
11 | simpr 479 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅) | |
12 | ordunifi 8500 | . . . 4 ⊢ ((𝑆 ⊆ On ∧ 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ 𝑆) | |
13 | 9, 10, 11, 12 | syl3anc 1439 | . . 3 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ 𝑆) |
14 | 7, 13 | sseldd 3822 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ ω) |
15 | 6, 14 | pm2.61dane 3057 | 1 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ⊆ wss 3792 ∅c0 4141 ∪ cuni 4673 Oncon0 5978 ωcom 7345 Fincfn 8243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-om 7346 df-1o 7845 df-er 8028 df-en 8244 df-fin 8247 |
This theorem is referenced by: ackbij1lem16 9394 isf32lem5 9516 finxpreclem4 33829 |
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