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Mirrors > Home > MPE Home > Th. List > card1 | Structured version Visualization version GIF version |
Description: A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.) |
Ref | Expression |
---|---|
card1 | ⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8255 | . . . . . . . 8 ⊢ 1o ∈ ω | |
2 | cardnn 9381 | . . . . . . . 8 ⊢ (1o ∈ ω → (card‘1o) = 1o) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ (card‘1o) = 1o |
4 | 1n0 8110 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
5 | 3, 4 | eqnetri 3086 | . . . . . 6 ⊢ (card‘1o) ≠ ∅ |
6 | carden2a 9384 | . . . . . 6 ⊢ (((card‘1o) = (card‘𝐴) ∧ (card‘1o) ≠ ∅) → 1o ≈ 𝐴) | |
7 | 5, 6 | mpan2 687 | . . . . 5 ⊢ ((card‘1o) = (card‘𝐴) → 1o ≈ 𝐴) |
8 | 7 | eqcoms 2829 | . . . 4 ⊢ ((card‘𝐴) = (card‘1o) → 1o ≈ 𝐴) |
9 | 8 | ensymd 8549 | . . 3 ⊢ ((card‘𝐴) = (card‘1o) → 𝐴 ≈ 1o) |
10 | carden2b 9385 | . . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o)) | |
11 | 9, 10 | impbii 210 | . 2 ⊢ ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o) |
12 | 3 | eqeq2i 2834 | . 2 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o) |
13 | en1 8565 | . 2 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
14 | 11, 12, 13 | 3bitr3i 302 | 1 ⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3016 ∅c0 4290 {csn 4559 class class class wbr 5058 ‘cfv 6349 ωcom 7568 1oc1o 8086 ≈ cen 8495 cardccrd 9353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7569 df-1o 8093 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-card 9357 |
This theorem is referenced by: cardsn 9387 |
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