MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sdom2en01 Structured version   Visualization version   GIF version

Theorem sdom2en01 10284
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 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))

Proof of Theorem sdom2en01
StepHypRef Expression
1 onfin2 9219 . . . . 5 ω = (On ∩ Fin)
2 inss2 4227 . . . . 5 (On ∩ Fin) ⊆ Fin
31, 2eqsstri 4014 . . . 4 ω ⊆ Fin
4 2onn 8629 . . . 4 2o ∈ ω
53, 4sselii 3977 . . 3 2o ∈ Fin
6 sdomdom 8964 . . 3 (𝐴 ≺ 2o𝐴 ≼ 2o)
7 domfi 9180 . . 3 ((2o ∈ Fin ∧ 𝐴 ≼ 2o) → 𝐴 ∈ Fin)
85, 6, 7sylancr 588 . 2 (𝐴 ≺ 2o𝐴 ∈ Fin)
9 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
10 0fin 9159 . . . 4 ∅ ∈ Fin
119, 10eqeltrdi 2842 . . 3 (𝐴 = ∅ → 𝐴 ∈ Fin)
12 1onn 8627 . . . . 5 1o ∈ ω
133, 12sselii 3977 . . . 4 1o ∈ Fin
14 enfi 9178 . . . 4 (𝐴 ≈ 1o → (𝐴 ∈ Fin ↔ 1o ∈ Fin))
1513, 14mpbiri 258 . . 3 (𝐴 ≈ 1o𝐴 ∈ Fin)
1611, 15jaoi 856 . 2 ((𝐴 = ∅ ∨ 𝐴 ≈ 1o) → 𝐴 ∈ Fin)
17 df2o3 8461 . . . . . 6 2o = {∅, 1o}
1817eleq2i 2826 . . . . 5 ((card‘𝐴) ∈ 2o ↔ (card‘𝐴) ∈ {∅, 1o})
19 fvex 6894 . . . . . 6 (card‘𝐴) ∈ V
2019elpr 4647 . . . . 5 ((card‘𝐴) ∈ {∅, 1o} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
2118, 20bitri 275 . . . 4 ((card‘𝐴) ∈ 2o ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
2221a1i 11 . . 3 (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o)))
23 cardnn 9945 . . . . . 6 (2o ∈ ω → (card‘2o) = 2o)
244, 23ax-mp 5 . . . . 5 (card‘2o) = 2o
2524eleq2i 2826 . . . 4 ((card‘𝐴) ∈ (card‘2o) ↔ (card‘𝐴) ∈ 2o)
26 finnum 9930 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27 2on 8467 . . . . . 6 2o ∈ On
28 onenon 9931 . . . . . 6 (2o ∈ On → 2o ∈ dom card)
2927, 28ax-mp 5 . . . . 5 2o ∈ dom card
30 cardsdom2 9970 . . . . 5 ((𝐴 ∈ dom card ∧ 2o ∈ dom card) → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
3126, 29, 30sylancl 587 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
3225, 31bitr3id 285 . . 3 (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o𝐴 ≺ 2o))
33 cardnueq0 9946 . . . . 5 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
3426, 33syl 17 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35 cardnn 9945 . . . . . . 7 (1o ∈ ω → (card‘1o) = 1o)
3612, 35ax-mp 5 . . . . . 6 (card‘1o) = 1o
3736eqeq2i 2746 . . . . 5 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
38 finnum 9930 . . . . . . 7 (1o ∈ Fin → 1o ∈ dom card)
3913, 38ax-mp 5 . . . . . 6 1o ∈ dom card
40 carden2 9969 . . . . . 6 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
4126, 39, 40sylancl 587 . . . . 5 (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
4237, 41bitr3id 285 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) = 1o𝐴 ≈ 1o))
4334, 42orbi12d 918 . . 3 (𝐴 ∈ Fin → (((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o) ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
4422, 32, 433bitr3d 309 . 2 (𝐴 ∈ Fin → (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
458, 16, 44pm5.21nii 380 1 (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 846   = wceq 1542  wcel 2107  cin 3945  c0 4320  {cpr 4626   class class class wbr 5144  dom cdm 5672  Oncon0 6356  cfv 6535  ωcom 7842  1oc1o 8446  2oc2o 8447  cen 8924  cdom 8925  csdm 8926  Fincfn 8927  cardccrd 9917
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 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
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 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-om 7843  df-1o 8453  df-2o 8454  df-er 8691  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-card 9921
This theorem is referenced by:  fin56  10375  en2top  22457
  Copyright terms: Public domain W3C validator