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Theorem card1 8832
 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 7764 . . . . . . . 8 1𝑜 ∈ ω
2 cardnn 8827 . . . . . . . 8 (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
31, 2ax-mp 5 . . . . . . 7 (card‘1𝑜) = 1𝑜
4 1n0 7620 . . . . . . 7 1𝑜 ≠ ∅
53, 4eqnetri 2893 . . . . . 6 (card‘1𝑜) ≠ ∅
6 carden2a 8830 . . . . . 6 (((card‘1𝑜) = (card‘𝐴) ∧ (card‘1𝑜) ≠ ∅) → 1𝑜𝐴)
75, 6mpan2 707 . . . . 5 ((card‘1𝑜) = (card‘𝐴) → 1𝑜𝐴)
87eqcoms 2659 . . . 4 ((card‘𝐴) = (card‘1𝑜) → 1𝑜𝐴)
98ensymd 8048 . . 3 ((card‘𝐴) = (card‘1𝑜) → 𝐴 ≈ 1𝑜)
10 carden2b 8831 . . 3 (𝐴 ≈ 1𝑜 → (card‘𝐴) = (card‘1𝑜))
119, 10impbii 199 . 2 ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜)
123eqeq2i 2663 . 2 ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
13 en1 8064 . 2 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
1411, 12, 133bitr3i 290 1 ((card‘𝐴) = 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823  ∅c0 3948  {csn 4210   class class class wbr 4685  ‘cfv 5926  ωcom 7107  1𝑜c1o 7598   ≈ cen 7994  cardccrd 8799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803 This theorem is referenced by:  cardsn  8833
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