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Theorem card1 9386
Description: A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)
Assertion
Ref Expression
card1 ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem card1
StepHypRef Expression
1 1onn 8255 . . . . . . . 8 1o ∈ ω
2 cardnn 9381 . . . . . . . 8 (1o ∈ ω → (card‘1o) = 1o)
31, 2ax-mp 5 . . . . . . 7 (card‘1o) = 1o
4 1n0 8110 . . . . . . 7 1o ≠ ∅
53, 4eqnetri 3086 . . . . . 6 (card‘1o) ≠ ∅
6 carden2a 9384 . . . . . 6 (((card‘1o) = (card‘𝐴) ∧ (card‘1o) ≠ ∅) → 1o𝐴)
75, 6mpan2 687 . . . . 5 ((card‘1o) = (card‘𝐴) → 1o𝐴)
87eqcoms 2829 . . . 4 ((card‘𝐴) = (card‘1o) → 1o𝐴)
98ensymd 8549 . . 3 ((card‘𝐴) = (card‘1o) → 𝐴 ≈ 1o)
10 carden2b 9385 . . 3 (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
119, 10impbii 210 . 2 ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)
123eqeq2i 2834 . 2 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
13 en1 8565 . 2 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
1411, 12, 133bitr3i 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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