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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 6244 | . . 3 ⊢ ∅ ∈ On | |
2 | cardonle 9538 | . . 3 ⊢ (∅ ∈ On → (card‘∅) ⊆ ∅) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (card‘∅) ⊆ ∅ |
4 | ss0b 4298 | . 2 ⊢ ((card‘∅) ⊆ ∅ ↔ (card‘∅) = ∅) | |
5 | 3, 4 | mpbi 233 | 1 ⊢ (card‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ∅c0 4223 Oncon0 6191 ‘cfv 6358 cardccrd 9516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-en 8605 df-card 9520 |
This theorem is referenced by: cardidm 9540 cardnueq0 9545 alephcard 9649 ackbij2lem2 9819 cf0 9830 cardcf 9831 cardeq0 10131 |
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