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Theorem sdom2en01 10297
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 9231 . . . . 5 Ο‰ = (On ∩ Fin)
2 inss2 4230 . . . . 5 (On ∩ Fin) βŠ† Fin
31, 2eqsstri 4017 . . . 4 Ο‰ βŠ† Fin
4 2onn 8641 . . . 4 2o ∈ Ο‰
53, 4sselii 3980 . . 3 2o ∈ Fin
6 sdomdom 8976 . . 3 (𝐴 β‰Ί 2o β†’ 𝐴 β‰Ό 2o)
7 domfi 9192 . . 3 ((2o ∈ Fin ∧ 𝐴 β‰Ό 2o) β†’ 𝐴 ∈ Fin)
85, 6, 7sylancr 588 . 2 (𝐴 β‰Ί 2o β†’ 𝐴 ∈ Fin)
9 id 22 . . . 4 (𝐴 = βˆ… β†’ 𝐴 = βˆ…)
10 0fin 9171 . . . 4 βˆ… ∈ Fin
119, 10eqeltrdi 2842 . . 3 (𝐴 = βˆ… β†’ 𝐴 ∈ Fin)
12 1onn 8639 . . . . 5 1o ∈ Ο‰
133, 12sselii 3980 . . . 4 1o ∈ Fin
14 enfi 9190 . . . 4 (𝐴 β‰ˆ 1o β†’ (𝐴 ∈ Fin ↔ 1o ∈ Fin))
1513, 14mpbiri 258 . . 3 (𝐴 β‰ˆ 1o β†’ 𝐴 ∈ Fin)
1611, 15jaoi 856 . 2 ((𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o) β†’ 𝐴 ∈ Fin)
17 df2o3 8474 . . . . . 6 2o = {βˆ…, 1o}
1817eleq2i 2826 . . . . 5 ((cardβ€˜π΄) ∈ 2o ↔ (cardβ€˜π΄) ∈ {βˆ…, 1o})
19 fvex 6905 . . . . . 6 (cardβ€˜π΄) ∈ V
2019elpr 4652 . . . . 5 ((cardβ€˜π΄) ∈ {βˆ…, 1o} ↔ ((cardβ€˜π΄) = βˆ… ∨ (cardβ€˜π΄) = 1o))
2118, 20bitri 275 . . . 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 9958 . . . . . 6 (2o ∈ Ο‰ β†’ (cardβ€˜2o) = 2o)
244, 23ax-mp 5 . . . . 5 (cardβ€˜2o) = 2o
2524eleq2i 2826 . . . 4 ((cardβ€˜π΄) ∈ (cardβ€˜2o) ↔ (cardβ€˜π΄) ∈ 2o)
26 finnum 9943 . . . . 5 (𝐴 ∈ Fin β†’ 𝐴 ∈ dom card)
27 2on 8480 . . . . . 6 2o ∈ On
28 onenon 9944 . . . . . 6 (2o ∈ On β†’ 2o ∈ dom card)
2927, 28ax-mp 5 . . . . 5 2o ∈ dom card
30 cardsdom2 9983 . . . . 5 ((𝐴 ∈ dom card ∧ 2o ∈ dom card) β†’ ((cardβ€˜π΄) ∈ (cardβ€˜2o) ↔ 𝐴 β‰Ί 2o))
3126, 29, 30sylancl 587 . . . 4 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) ∈ (cardβ€˜2o) ↔ 𝐴 β‰Ί 2o))
3225, 31bitr3id 285 . . 3 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) ∈ 2o ↔ 𝐴 β‰Ί 2o))
33 cardnueq0 9959 . . . . 5 (𝐴 ∈ dom card β†’ ((cardβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
3426, 33syl 17 . . . 4 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
35 cardnn 9958 . . . . . . 7 (1o ∈ Ο‰ β†’ (cardβ€˜1o) = 1o)
3612, 35ax-mp 5 . . . . . 6 (cardβ€˜1o) = 1o
3736eqeq2i 2746 . . . . 5 ((cardβ€˜π΄) = (cardβ€˜1o) ↔ (cardβ€˜π΄) = 1o)
38 finnum 9943 . . . . . . 7 (1o ∈ Fin β†’ 1o ∈ dom card)
3913, 38ax-mp 5 . . . . . 6 1o ∈ dom card
40 carden2 9982 . . . . . 6 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
4126, 39, 40sylancl 587 . . . . 5 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) = (cardβ€˜1o) ↔ 𝐴 β‰ˆ 1o))
4237, 41bitr3id 285 . . . 4 (𝐴 ∈ Fin β†’ ((cardβ€˜π΄) = 1o ↔ 𝐴 β‰ˆ 1o))
4334, 42orbi12d 918 . . 3 (𝐴 ∈ Fin β†’ (((cardβ€˜π΄) = βˆ… ∨ (cardβ€˜π΄) = 1o) ↔ (𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o)))
4422, 32, 433bitr3d 309 . 2 (𝐴 ∈ Fin β†’ (𝐴 β‰Ί 2o ↔ (𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o)))
458, 16, 44pm5.21nii 380 1 (𝐴 β‰Ί 2o ↔ (𝐴 = βˆ… ∨ 𝐴 β‰ˆ 1o))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∨ wo 846   = wceq 1542   ∈ wcel 2107   ∩ cin 3948  βˆ…c0 4323  {cpr 4631   class class class wbr 5149  dom cdm 5677  Oncon0 6365  β€˜cfv 6544  Ο‰com 7855  1oc1o 8459  2oc2o 8460   β‰ˆ cen 8936   β‰Ό cdom 8937   β‰Ί csdm 8938  Fincfn 8939  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-2o 8467  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934
This theorem is referenced by:  fin56  10388  en2top  22488
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