Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fiunicl | Structured version Visualization version GIF version |
Description: If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
fiunicl.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝐴) |
fiunicl.2 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fiunicl.3 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Ref | Expression |
---|---|
fiunicl | ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniiun 4973 | . 2 ⊢ ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 𝑧 | |
2 | nfv 1906 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴) | |
4 | fiunicl.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝐴) | |
5 | fiunicl.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | fiunicl.3 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
7 | 2, 3, 4, 5, 6 | fiiuncl 41204 | . 2 ⊢ (𝜑 → ∪ 𝑧 ∈ 𝐴 𝑧 ∈ 𝐴) |
8 | 1, 7 | eqeltrid 2914 | 1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 ∈ wcel 2105 ≠ wne 3013 ∪ cun 3931 ∅c0 4288 ∪ cuni 4830 ∪ ciun 4910 Fincfn 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-1o 8091 df-er 8278 df-en 8498 df-fin 8501 |
This theorem is referenced by: (None) |
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