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Theorem card1 9991
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 8660 . . . . . . . 8 1o ∈ ω
2 cardnn 9986 . . . . . . . 8 (1o ∈ ω → (card‘1o) = 1o)
31, 2ax-mp 5 . . . . . . 7 (card‘1o) = 1o
4 1n0 8508 . . . . . . 7 1o ≠ ∅
53, 4eqnetri 3008 . . . . . 6 (card‘1o) ≠ ∅
6 carden2a 9989 . . . . . 6 (((card‘1o) = (card‘𝐴) ∧ (card‘1o) ≠ ∅) → 1o𝐴)
75, 6mpan2 690 . . . . 5 ((card‘1o) = (card‘𝐴) → 1o𝐴)
87eqcoms 2736 . . . 4 ((card‘𝐴) = (card‘1o) → 1o𝐴)
98ensymd 9025 . . 3 ((card‘𝐴) = (card‘1o) → 𝐴 ≈ 1o)
10 carden2b 9990 . . 3 (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
119, 10impbii 208 . 2 ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)
123eqeq2i 2741 . 2 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
13 en1 9045 . 2 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
1411, 12, 133bitr3i 301 1 ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wex 1774  wcel 2099  wne 2937  c0 4323  {csn 4629   class class class wbr 5148  cfv 6548  ωcom 7870  1oc1o 8479  cen 8960  cardccrd 9958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-om 7871  df-1o 8486  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-card 9962
This theorem is referenced by:  cardsn  9992
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