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Mirrors > Home > MPE Home > Th. List > fieq0 | Structured version Visualization version GIF version |
Description: A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
fieq0 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6664 | . . 3 ⊢ (𝐴 = ∅ → (fi‘𝐴) = (fi‘∅)) | |
2 | fi0 8878 | . . 3 ⊢ (fi‘∅) = ∅ | |
3 | 1, 2 | syl6eq 2872 | . 2 ⊢ (𝐴 = ∅ → (fi‘𝐴) = ∅) |
4 | ssfii 8877 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | |
5 | sseq0 4352 | . . . 4 ⊢ ((𝐴 ⊆ (fi‘𝐴) ∧ (fi‘𝐴) = ∅) → 𝐴 = ∅) | |
6 | 4, 5 | sylan 582 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (fi‘𝐴) = ∅) → 𝐴 = ∅) |
7 | 6 | ex 415 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((fi‘𝐴) = ∅ → 𝐴 = ∅)) |
8 | 3, 7 | impbid2 228 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∅c0 4290 ‘cfv 6349 ficfi 8868 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-1o 8096 df-en 8504 df-fin 8507 df-fi 8869 |
This theorem is referenced by: fsubbas 22469 |
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