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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 4916 | . . . 4 ⊢ (𝑆 = ∅ → ∪ 𝑆 = ∪ ∅) | |
2 | uni0 4935 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | peano1 7892 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | eqeltri 2822 | . . . 4 ⊢ ∪ ∅ ∈ ω |
5 | 1, 4 | eqeltrdi 2834 | . . 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 7872 | . . . . 5 ⊢ ω ⊆ On | |
9 | 7, 8 | sstrdi 3991 | . . . 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 9320 | . . . 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 3979 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ ω) |
15 | 6, 14 | pm2.61dane 3019 | 1 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ⊆ wss 3946 ∅c0 4322 ∪ cuni 4905 Oncon0 6368 ωcom 7868 Fincfn 8966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-om 7869 df-en 8967 df-fin 8970 |
This theorem is referenced by: ackbij1lem16 10269 isf32lem5 10391 finxpreclem4 37114 |
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