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Theorem sdom2en01 10299
Description: A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Assertion
Ref Expression
sdom2en01 (𝐴 β‰Ί 2o ↔ (𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o))

Proof of Theorem sdom2en01
StepHypRef Expression
1 onfin2 9233 . . . . 5 Ο‰ = (On ∩ Fin)
2 inss2 4229 . . . . 5 (On ∩ Fin) βŠ† Fin
31, 2eqsstri 4016 . . . 4 Ο‰ βŠ† Fin
4 2onn 8643 . . . 4 2o ∈ Ο‰
53, 4sselii 3979 . . 3 2o ∈ Fin
6 sdomdom 8978 . . 3 (𝐴 β‰Ί 2o β†’ 𝐴 β‰Ό 2o)
7 domfi 9194 . . 3 ((2o ∈ Fin ∧ 𝐴 β‰Ό 2o) β†’ 𝐴 ∈ Fin)
85, 6, 7sylancr 587 . 2 (𝐴 β‰Ί 2o β†’ 𝐴 ∈ Fin)
9 id 22 . . . 4 (𝐴 = βˆ… β†’ 𝐴 = βˆ…)
10 0fin 9173 . . . 4 βˆ… ∈ Fin
119, 10eqeltrdi 2841 . . 3 (𝐴 = βˆ… β†’ 𝐴 ∈ Fin)
12 1onn 8641 . . . . 5 1o ∈ Ο‰
133, 12sselii 3979 . . . 4 1o ∈ Fin
14 enfi 9192 . . . 4 (𝐴 β‰ˆ 1o β†’ (𝐴 ∈ Fin ↔ 1o ∈ Fin))
1513, 14mpbiri 257 . . 3 (𝐴 β‰ˆ 1o β†’ 𝐴 ∈ Fin)
1611, 15jaoi 855 . 2 ((𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o) β†’ 𝐴 ∈ Fin)
17 df2o3 8476 . . . . . 6 2o = {βˆ…, 1o}
1817eleq2i 2825 . . . . 5 ((cardβ€˜π΄) ∈ 2o ↔ (cardβ€˜π΄) ∈ {βˆ…, 1o})
19 fvex 6904 . . . . . 6 (cardβ€˜π΄) ∈ V
2019elpr 4651 . . . . 5 ((cardβ€˜π΄) ∈ {βˆ…, 1o} ↔ ((cardβ€˜π΄) = βˆ… ∨ (cardβ€˜π΄) = 1o))
2118, 20bitri 274 . . . 4 ((cardβ€˜π΄) ∈ 2o ↔ ((cardβ€˜π΄) = βˆ… ∨ (cardβ€˜π΄) = 1o))
2221a1i 11 . . 3 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) ∈ 2o ↔ ((cardβ€˜π΄) = βˆ… ∨ (cardβ€˜π΄) = 1o)))
23 cardnn 9960 . . . . . 6 (2o ∈ Ο‰ β†’ (cardβ€˜2o) = 2o)
244, 23ax-mp 5 . . . . 5 (cardβ€˜2o) = 2o
2524eleq2i 2825 . . . 4 ((cardβ€˜π΄) ∈ (cardβ€˜2o) ↔ (cardβ€˜π΄) ∈ 2o)
26 finnum 9945 . . . . 5 (𝐴 ∈ Fin β†’ 𝐴 ∈ dom card)
27 2on 8482 . . . . . 6 2o ∈ On
28 onenon 9946 . . . . . 6 (2o ∈ On β†’ 2o ∈ dom card)
2927, 28ax-mp 5 . . . . 5 2o ∈ dom card
30 cardsdom2 9985 . . . . 5 ((𝐴 ∈ dom card ∧ 2o ∈ dom card) β†’ ((cardβ€˜π΄) ∈ (cardβ€˜2o) ↔ 𝐴 β‰Ί 2o))
3126, 29, 30sylancl 586 . . . 4 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) ∈ (cardβ€˜2o) ↔ 𝐴 β‰Ί 2o))
3225, 31bitr3id 284 . . 3 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) ∈ 2o ↔ 𝐴 β‰Ί 2o))
33 cardnueq0 9961 . . . . 5 (𝐴 ∈ dom card β†’ ((cardβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
3426, 33syl 17 . . . 4 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
35 cardnn 9960 . . . . . . 7 (1o ∈ Ο‰ β†’ (cardβ€˜1o) = 1o)
3612, 35ax-mp 5 . . . . . 6 (cardβ€˜1o) = 1o
3736eqeq2i 2745 . . . . 5 ((cardβ€˜π΄) = (cardβ€˜1o) ↔ (cardβ€˜π΄) = 1o)
38 finnum 9945 . . . . . . 7 (1o ∈ Fin β†’ 1o ∈ dom card)
3913, 38ax-mp 5 . . . . . 6 1o ∈ dom card
40 carden2 9984 . . . . . 6 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
4126, 39, 40sylancl 586 . . . . 5 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
4237, 41bitr3id 284 . . . 4 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) = 1o ↔ 𝐴 β‰ˆ 1o))
4334, 42orbi12d 917 . . 3 (𝐴 ∈ Fin β†’ (((cardβ€˜π΄) = βˆ… ∨ (cardβ€˜π΄) = 1o) ↔ (𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o)))
4422, 32, 433bitr3d 308 . 2 (𝐴 ∈ Fin β†’ (𝐴 β‰Ί 2o ↔ (𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o)))
458, 16, 44pm5.21nii 379 1 (𝐴 β‰Ί 2o ↔ (𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ∩ cin 3947  βˆ…c0 4322  {cpr 4630   class class class wbr 5148  dom cdm 5676  Oncon0 6364  β€˜cfv 6543  Ο‰com 7857  1oc1o 8461  2oc2o 8462   β‰ˆ cen 8938   β‰Ό cdom 8939   β‰Ί csdm 8940  Fincfn 8941  cardccrd 9932
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1o 8468  df-2o 8469  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936
This theorem is referenced by:  fin56  10390  en2top  22495
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