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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 9386 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 9343). (Contributed by NM, 11-Mar-2006.) |
Ref | Expression |
---|---|
unifi2 | ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ∪ 𝐴 ≺ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfinite2 9335 | . . 3 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
2 | isfinite2 9335 | . . . . 5 ⊢ (𝑥 ≺ ω → 𝑥 ∈ Fin) | |
3 | 2 | ralimi 3073 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≺ ω → ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin) |
4 | dfss3 3968 | . . . 4 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin) | |
5 | 3, 4 | sylibr 233 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≺ ω → 𝐴 ⊆ Fin) |
6 | unifi 9386 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin) | |
7 | 1, 5, 6 | syl2an 594 | . 2 ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ∪ 𝐴 ∈ Fin) |
8 | fin2inf 9343 | . . . 4 ⊢ (𝐴 ≺ ω → ω ∈ V) | |
9 | 8 | adantr 479 | . . 3 ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ω ∈ V) |
10 | isfiniteg 9338 | . . 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 231 | 1 ⊢ ((𝐴 ≺ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ ω) → ∪ 𝐴 ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ⊆ wss 3947 ∪ cuni 4913 class class class wbr 5153 ωcom 7876 ≺ csdm 8973 Fincfn 8974 |
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 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 |
This theorem is referenced by: (None) |
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