Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > unifi3 | Structured version Visualization version GIF version |
Description: If a union is finite, then all its elements are finite. See unifi 8807. (Contributed by Thierry Arnoux, 27-Aug-2017.) |
Ref | Expression |
---|---|
unifi3 | ⊢ (∪ 𝐴 ∈ Fin → 𝐴 ⊆ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 4861 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) | |
2 | ssfi 8732 | . . . 4 ⊢ ((∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴) → 𝑥 ∈ Fin) | |
3 | 2 | ex 415 | . . 3 ⊢ (∪ 𝐴 ∈ Fin → (𝑥 ⊆ ∪ 𝐴 → 𝑥 ∈ Fin)) |
4 | 1, 3 | syl5 34 | . 2 ⊢ (∪ 𝐴 ∈ Fin → (𝑥 ∈ 𝐴 → 𝑥 ∈ Fin)) |
5 | 4 | ssrdv 3973 | 1 ⊢ (∪ 𝐴 ∈ Fin → 𝐴 ⊆ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3936 ∪ cuni 4832 Fincfn 8503 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
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-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-om 7575 df-er 8283 df-en 8504 df-fin 8507 |
This theorem is referenced by: fpwrelmapffslem 30462 |
Copyright terms: Public domain | W3C validator |