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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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