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Mirrors > Home > ILE Home > Th. List > card0 | GIF version |
Description: The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
Ref | Expression |
---|---|
card0 | ⊢ (card‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 4417 | . . 3 ⊢ ∅ ∈ On | |
2 | cardonle 7233 | . . 3 ⊢ (∅ ∈ On → (card‘∅) ⊆ ∅) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (card‘∅) ⊆ ∅ |
4 | ss0b 3482 | . 2 ⊢ ((card‘∅) ⊆ ∅ ↔ (card‘∅) = ∅) | |
5 | 3, 4 | mpbi 145 | 1 ⊢ (card‘∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 ⊆ wss 3149 ∅c0 3442 Oncon0 4388 ‘cfv 5242 cardccrd 7225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-id 4318 df-iord 4391 df-on 4393 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-en 6782 df-card 7226 |
This theorem is referenced by: (None) |
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