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Mirrors > Home > MPE Home > Th. List > fieq0 | Structured version Visualization version GIF version |
Description: If 𝐴 is not empty, the class of all the finite intersections of 𝐴 is not empty either. (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 6411 | . . 3 ⊢ (𝐴 = ∅ → (fi‘𝐴) = (fi‘∅)) | |
2 | fi0 8568 | . . 3 ⊢ (fi‘∅) = ∅ | |
3 | 1, 2 | syl6eq 2849 | . 2 ⊢ (𝐴 = ∅ → (fi‘𝐴) = ∅) |
4 | ssfii 8567 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | |
5 | sseq0 4171 | . . . 4 ⊢ ((𝐴 ⊆ (fi‘𝐴) ∧ (fi‘𝐴) = ∅) → 𝐴 = ∅) | |
6 | 4, 5 | sylan 576 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (fi‘𝐴) = ∅) → 𝐴 = ∅) |
7 | 6 | ex 402 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((fi‘𝐴) = ∅ → 𝐴 = ∅)) |
8 | 3, 7 | impbid2 218 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ⊆ wss 3769 ∅c0 4115 ‘cfv 6101 ficfi 8558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-om 7300 df-1o 7799 df-en 8196 df-fin 8199 df-fi 8559 |
This theorem is referenced by: fsubbas 21999 |
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