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Mirrors > Home > MPE Home > Th. List > cardeq0 | Structured version Visualization version GIF version |
Description: Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.) |
Ref | Expression |
---|---|
cardeq0 | ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5028 | . . 3 ⊢ ∅ ∈ V | |
2 | carden 9710 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → ((card‘𝐴) = (card‘∅) ↔ 𝐴 ≈ ∅)) | |
3 | 1, 2 | mpan2 681 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = (card‘∅) ↔ 𝐴 ≈ ∅)) |
4 | card0 9119 | . . 3 ⊢ (card‘∅) = ∅ | |
5 | 4 | eqeq2i 2790 | . 2 ⊢ ((card‘𝐴) = (card‘∅) ↔ (card‘𝐴) = ∅) |
6 | en0 8306 | . 2 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
7 | 3, 5, 6 | 3bitr3g 305 | 1 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 class class class wbr 4888 ‘cfv 6137 ≈ cen 8240 cardccrd 9096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-ac2 9622 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-wrecs 7691 df-recs 7753 df-er 8028 df-en 8244 df-card 9100 df-ac 9274 |
This theorem is referenced by: tskcard 9940 |
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