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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 6897 | . . 3 ⊢ (𝐴 = ∅ → (fi‘𝐴) = (fi‘∅)) | |
2 | fi0 9444 | . . 3 ⊢ (fi‘∅) = ∅ | |
3 | 1, 2 | eqtrdi 2784 | . 2 ⊢ (𝐴 = ∅ → (fi‘𝐴) = ∅) |
4 | ssfii 9443 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | |
5 | sseq0 4400 | . . . 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 1534 ∈ wcel 2099 ⊆ wss 3947 ∅c0 4323 ‘cfv 6548 ficfi 9434 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-om 7871 df-1o 8487 df-en 8965 df-fin 8968 df-fi 9435 |
This theorem is referenced by: fsubbas 23784 |
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