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Mirrors > Home > MPE Home > Th. List > cardnueq0 | Structured version Visualization version GIF version |
Description: The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.) |
Ref | Expression |
---|---|
cardnueq0 | ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardid2 9455 | . . . 4 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
2 | 1 | ensymd 8606 | . . 3 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴)) |
3 | breq2 5034 | . . . 4 ⊢ ((card‘𝐴) = ∅ → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ ∅)) | |
4 | en0 8618 | . . . 4 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | |
5 | 3, 4 | bitrdi 290 | . . 3 ⊢ ((card‘𝐴) = ∅ → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 = ∅)) |
6 | 2, 5 | syl5ibcom 248 | . 2 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ → 𝐴 = ∅)) |
7 | fveq2 6674 | . . 3 ⊢ (𝐴 = ∅ → (card‘𝐴) = (card‘∅)) | |
8 | card0 9460 | . . 3 ⊢ (card‘∅) = ∅ | |
9 | 7, 8 | eqtrdi 2789 | . 2 ⊢ (𝐴 = ∅ → (card‘𝐴) = ∅) |
10 | 6, 9 | impbid1 228 | 1 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2114 ∅c0 4211 class class class wbr 5030 dom cdm 5525 ‘cfv 6339 ≈ cen 8552 cardccrd 9437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ord 6175 df-on 6176 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-er 8320 df-en 8556 df-card 9441 |
This theorem is referenced by: carddomi2 9472 cfeq0 9756 cfsuc 9757 sdom2en01 9802 |
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