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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 4444 | . . . 4 ⊢ (𝑆 = ∅ → ∪ 𝑆 = ∪ ∅) | |
2 | uni0 4465 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | peano1 7085 | . . . . 5 ⊢ ∅ ∈ ω | |
4 | 2, 3 | eqeltri 2697 | . . . 4 ⊢ ∪ ∅ ∈ ω |
5 | 1, 4 | syl6eqel 2709 | . . 3 ⊢ (𝑆 = ∅ → ∪ 𝑆 ∈ ω) |
6 | 5 | adantl 482 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 = ∅) → ∪ 𝑆 ∈ ω) |
7 | simpll 790 | . . 3 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ω) | |
8 | omsson 7069 | . . . . 5 ⊢ ω ⊆ On | |
9 | 7, 8 | syl6ss 3615 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ On) |
10 | simplr 792 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ Fin) | |
11 | simpr 477 | . . . 4 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅) | |
12 | ordunifi 8210 | . . . 4 ⊢ ((𝑆 ⊆ On ∧ 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ 𝑆) | |
13 | 9, 10, 11, 12 | syl3anc 1326 | . . 3 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ 𝑆) |
14 | 7, 13 | sseldd 3604 | . 2 ⊢ (((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) ∧ 𝑆 ≠ ∅) → ∪ 𝑆 ∈ ω) |
15 | 6, 14 | pm2.61dane 2881 | 1 ⊢ ((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → ∪ 𝑆 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 Oncon0 5723 ωcom 7065 Fincfn 7955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-er 7742 df-en 7956 df-fin 7959 |
This theorem is referenced by: ackbij1lem16 9057 isf32lem5 9179 finxpreclem4 33231 |
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