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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 4913 | . . . 4 ⊢ (𝑆 = ∅ → ∪ 𝑆 = ∪ ∅) | |
2 | uni0 4932 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | peano1 7875 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | eqeltri 2823 | . . . 4 ⊢ ∪ ∅ ∈ ω |
5 | 1, 4 | eqeltrdi 2835 | . . 3 ⊢ (𝑆 = ∅ → ∪ 𝑆 ∈ ω) |
6 | 5 | adantl 481 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 = ∅) → ∪ 𝑆 ∈ ω) |
7 | simpll 764 | . . 3 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ω) | |
8 | omsson 7855 | . . . . 5 ⊢ ω ⊆ On | |
9 | 7, 8 | sstrdi 3989 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ On) |
10 | simplr 766 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ Fin) | |
11 | simpr 484 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅) | |
12 | ordunifi 9292 | . . . 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 3978 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ ω) |
15 | 6, 14 | pm2.61dane 3023 | 1 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ⊆ wss 3943 ∅c0 4317 ∪ cuni 4902 Oncon0 6357 ωcom 7851 Fincfn 8938 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-om 7852 df-en 8939 df-fin 8942 |
This theorem is referenced by: ackbij1lem16 10229 isf32lem5 10351 finxpreclem4 36781 |
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