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Theorem card1 9373
 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 8240 . . . . . . . 8 1o ∈ ω
2 cardnn 9368 . . . . . . . 8 (1o ∈ ω → (card‘1o) = 1o)
31, 2ax-mp 5 . . . . . . 7 (card‘1o) = 1o
4 1n0 8094 . . . . . . 7 1o ≠ ∅
53, 4eqnetri 3077 . . . . . 6 (card‘1o) ≠ ∅
6 carden2a 9371 . . . . . 6 (((card‘1o) = (card‘𝐴) ∧ (card‘1o) ≠ ∅) → 1o𝐴)
75, 6mpan2 690 . . . . 5 ((card‘1o) = (card‘𝐴) → 1o𝐴)
87eqcoms 2829 . . . 4 ((card‘𝐴) = (card‘1o) → 1o𝐴)
98ensymd 8535 . . 3 ((card‘𝐴) = (card‘1o) → 𝐴 ≈ 1o)
10 carden2b 9372 . . 3 (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o))
119, 10impbii 212 . 2 ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o)
123eqeq2i 2834 . 2 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
13 en1 8551 . 2 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
1411, 12, 133bitr3i 304 1 ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ≠ wne 3007  ∅c0 4266  {csn 4540   class class class wbr 5039  ‘cfv 6328  ωcom 7555  1oc1o 8070   ≈ cen 8481  cardccrd 9340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7556  df-1o 8077  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-card 9344 This theorem is referenced by:  cardsn  9374
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