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Theorem sdom2en01 9162
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 8193 . . . . 5 ω = (On ∩ Fin)
2 inss2 3867 . . . . 5 (On ∩ Fin) ⊆ Fin
31, 2eqsstri 3668 . . . 4 ω ⊆ Fin
4 2onn 7765 . . . 4 2𝑜 ∈ ω
53, 4sselii 3633 . . 3 2𝑜 ∈ Fin
6 sdomdom 8025 . . 3 (𝐴 ≺ 2𝑜𝐴 ≼ 2𝑜)
7 domfi 8222 . . 3 ((2𝑜 ∈ Fin ∧ 𝐴 ≼ 2𝑜) → 𝐴 ∈ Fin)
85, 6, 7sylancr 696 . 2 (𝐴 ≺ 2𝑜𝐴 ∈ Fin)
9 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
10 0fin 8229 . . . 4 ∅ ∈ Fin
119, 10syl6eqel 2738 . . 3 (𝐴 = ∅ → 𝐴 ∈ Fin)
12 1onn 7764 . . . . 5 1𝑜 ∈ ω
133, 12sselii 3633 . . . 4 1𝑜 ∈ Fin
14 enfi 8217 . . . 4 (𝐴 ≈ 1𝑜 → (𝐴 ∈ Fin ↔ 1𝑜 ∈ Fin))
1513, 14mpbiri 248 . . 3 (𝐴 ≈ 1𝑜𝐴 ∈ Fin)
1611, 15jaoi 393 . 2 ((𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜) → 𝐴 ∈ Fin)
17 df2o3 7618 . . . . . 6 2𝑜 = {∅, 1𝑜}
1817eleq2i 2722 . . . . 5 ((card‘𝐴) ∈ 2𝑜 ↔ (card‘𝐴) ∈ {∅, 1𝑜})
19 fvex 6239 . . . . . 6 (card‘𝐴) ∈ V
2019elpr 4231 . . . . 5 ((card‘𝐴) ∈ {∅, 1𝑜} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜))
2118, 20bitri 264 . . . 4 ((card‘𝐴) ∈ 2𝑜 ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜))
2221a1i 11 . . 3 (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2𝑜 ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜)))
23 cardnn 8827 . . . . . 6 (2𝑜 ∈ ω → (card‘2𝑜) = 2𝑜)
244, 23ax-mp 5 . . . . 5 (card‘2𝑜) = 2𝑜
2524eleq2i 2722 . . . 4 ((card‘𝐴) ∈ (card‘2𝑜) ↔ (card‘𝐴) ∈ 2𝑜)
26 finnum 8812 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27 2on 7613 . . . . . 6 2𝑜 ∈ On
28 onenon 8813 . . . . . 6 (2𝑜 ∈ On → 2𝑜 ∈ dom card)
2927, 28ax-mp 5 . . . . 5 2𝑜 ∈ dom card
30 cardsdom2 8852 . . . . 5 ((𝐴 ∈ dom card ∧ 2𝑜 ∈ dom card) → ((card‘𝐴) ∈ (card‘2𝑜) ↔ 𝐴 ≺ 2𝑜))
3126, 29, 30sylancl 695 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2𝑜) ↔ 𝐴 ≺ 2𝑜))
3225, 31syl5bbr 274 . . 3 (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2𝑜𝐴 ≺ 2𝑜))
33 cardnueq0 8828 . . . . 5 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
3426, 33syl 17 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35 cardnn 8827 . . . . . . 7 (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
3612, 35ax-mp 5 . . . . . 6 (card‘1𝑜) = 1𝑜
3736eqeq2i 2663 . . . . 5 ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
38 finnum 8812 . . . . . . 7 (1𝑜 ∈ Fin → 1𝑜 ∈ dom card)
3913, 38ax-mp 5 . . . . . 6 1𝑜 ∈ dom card
40 carden2 8851 . . . . . 6 ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
4126, 39, 40sylancl 695 . . . . 5 (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
4237, 41syl5bbr 274 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) = 1𝑜𝐴 ≈ 1𝑜))
4334, 42orbi12d 746 . . 3 (𝐴 ∈ Fin → (((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜) ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)))
4422, 32, 433bitr3d 298 . 2 (𝐴 ∈ Fin → (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)))
458, 16, 44pm5.21nii 367 1 (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1523  wcel 2030  cin 3606  c0 3948  {cpr 4212   class class class wbr 4685  dom cdm 5143  Oncon0 5761  cfv 5926  ωcom 7107  1𝑜c1o 7598  2𝑜c2o 7599  cen 7994  cdom 7995  csdm 7996  Fincfn 7997  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803
This theorem is referenced by:  fin56  9253  en2top  20837
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