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Mirrors > Home > MPE Home > Th. List > unifi2 | Structured version Visualization version GIF version |
Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 8812 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8780). (Contributed by NM, 11-Mar-2006.) |
Ref | Expression |
---|---|
unifi2 | ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ∪ 𝐴 ≺ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfinite2 8775 | . . 3 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
2 | isfinite2 8775 | . . . . 5 ⊢ (𝑥 ≺ ω → 𝑥 ∈ Fin) | |
3 | 2 | ralimi 3160 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≺ ω → ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin) |
4 | dfss3 3955 | . . . 4 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin) | |
5 | 3, 4 | sylibr 236 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≺ ω → 𝐴 ⊆ Fin) |
6 | unifi 8812 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin) | |
7 | 1, 5, 6 | syl2an 597 | . 2 ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ∪ 𝐴 ∈ Fin) |
8 | fin2inf 8780 | . . . 4 ⊢ (𝐴 ≺ ω → ω ∈ V) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ω ∈ V) |
10 | isfiniteg 8777 | . . 3 ⊢ (ω ∈ V → (∪ 𝐴 ∈ Fin ↔ ∪ 𝐴 ≺ ω)) | |
11 | 9, 10 | syl 17 | . 2 ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → (∪ 𝐴 ∈ Fin ↔ ∪ 𝐴 ≺ ω)) |
12 | 7, 11 | mpbid 234 | 1 ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ∪ 𝐴 ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 ∪ cuni 4837 class class class wbr 5065 ωcom 7579 ≺ csdm 8507 Fincfn 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 |
This theorem is referenced by: (None) |
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