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Mirrors > Home > MPE Home > Th. List > card0 | Structured version Visualization version 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 6230 | . . 3 ⊢ ∅ ∈ On | |
2 | cardonle 9372 | . . 3 ⊢ (∅ ∈ On → (card‘∅) ⊆ ∅) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (card‘∅) ⊆ ∅ |
4 | ss0b 4337 | . 2 ⊢ ((card‘∅) ⊆ ∅ ↔ (card‘∅) = ∅) | |
5 | 3, 4 | mpbi 232 | 1 ⊢ (card‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ⊆ wss 3924 ∅c0 4279 Oncon0 6177 ‘cfv 6341 cardccrd 9350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-ord 6180 df-on 6181 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-en 8496 df-card 9354 |
This theorem is referenced by: cardidm 9374 cardnueq0 9379 alephcard 9482 ackbij2lem2 9648 cf0 9659 cardcf 9660 cardeq0 9960 |
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