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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 6768 | . . 3 ⊢ (𝐴 = ∅ → (fi‘𝐴) = (fi‘∅)) | |
2 | fi0 9140 | . . 3 ⊢ (fi‘∅) = ∅ | |
3 | 1, 2 | eqtrdi 2795 | . 2 ⊢ (𝐴 = ∅ → (fi‘𝐴) = ∅) |
4 | ssfii 9139 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | |
5 | sseq0 4338 | . . . 4 ⊢ ((𝐴 ⊆ (fi‘𝐴) ∧ (fi‘𝐴) = ∅) → 𝐴 = ∅) | |
6 | 4, 5 | sylan 579 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (fi‘𝐴) = ∅) → 𝐴 = ∅) |
7 | 6 | ex 412 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((fi‘𝐴) = ∅ → 𝐴 = ∅)) |
8 | 3, 7 | impbid2 225 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2109 ⊆ wss 3891 ∅c0 4261 ‘cfv 6430 ficfi 9130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-om 7701 df-1o 8281 df-en 8708 df-fin 8711 df-fi 9131 |
This theorem is referenced by: fsubbas 22999 |
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