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Theorem sdom2en01 9412
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 8394 . . . . 5 ω = (On ∩ Fin)
2 inss2 4029 . . . . 5 (On ∩ Fin) ⊆ Fin
31, 2eqsstri 3831 . . . 4 ω ⊆ Fin
4 2onn 7960 . . . 4 2𝑜 ∈ ω
53, 4sselii 3795 . . 3 2𝑜 ∈ Fin
6 sdomdom 8223 . . 3 (𝐴 ≺ 2𝑜𝐴 ≼ 2𝑜)
7 domfi 8423 . . 3 ((2𝑜 ∈ Fin ∧ 𝐴 ≼ 2𝑜) → 𝐴 ∈ Fin)
85, 6, 7sylancr 582 . 2 (𝐴 ≺ 2𝑜𝐴 ∈ Fin)
9 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
10 0fin 8430 . . . 4 ∅ ∈ Fin
119, 10syl6eqel 2886 . . 3 (𝐴 = ∅ → 𝐴 ∈ Fin)
12 1onn 7959 . . . . 5 1𝑜 ∈ ω
133, 12sselii 3795 . . . 4 1𝑜 ∈ Fin
14 enfi 8418 . . . 4 (𝐴 ≈ 1𝑜 → (𝐴 ∈ Fin ↔ 1𝑜 ∈ Fin))
1513, 14mpbiri 250 . . 3 (𝐴 ≈ 1𝑜𝐴 ∈ Fin)
1611, 15jaoi 884 . 2 ((𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜) → 𝐴 ∈ Fin)
17 df2o3 7813 . . . . . 6 2𝑜 = {∅, 1𝑜}
1817eleq2i 2870 . . . . 5 ((card‘𝐴) ∈ 2𝑜 ↔ (card‘𝐴) ∈ {∅, 1𝑜})
19 fvex 6424 . . . . . 6 (card‘𝐴) ∈ V
2019elpr 4391 . . . . 5 ((card‘𝐴) ∈ {∅, 1𝑜} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜))
2118, 20bitri 267 . . . 4 ((card‘𝐴) ∈ 2𝑜 ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜))
2221a1i 11 . . 3 (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2𝑜 ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜)))
23 cardnn 9075 . . . . . 6 (2𝑜 ∈ ω → (card‘2𝑜) = 2𝑜)
244, 23ax-mp 5 . . . . 5 (card‘2𝑜) = 2𝑜
2524eleq2i 2870 . . . 4 ((card‘𝐴) ∈ (card‘2𝑜) ↔ (card‘𝐴) ∈ 2𝑜)
26 finnum 9060 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27 2on 7808 . . . . . 6 2𝑜 ∈ On
28 onenon 9061 . . . . . 6 (2𝑜 ∈ On → 2𝑜 ∈ dom card)
2927, 28ax-mp 5 . . . . 5 2𝑜 ∈ dom card
30 cardsdom2 9100 . . . . 5 ((𝐴 ∈ dom card ∧ 2𝑜 ∈ dom card) → ((card‘𝐴) ∈ (card‘2𝑜) ↔ 𝐴 ≺ 2𝑜))
3126, 29, 30sylancl 581 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2𝑜) ↔ 𝐴 ≺ 2𝑜))
3225, 31syl5bbr 277 . . 3 (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2𝑜𝐴 ≺ 2𝑜))
33 cardnueq0 9076 . . . . 5 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
3426, 33syl 17 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35 cardnn 9075 . . . . . . 7 (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
3612, 35ax-mp 5 . . . . . 6 (card‘1𝑜) = 1𝑜
3736eqeq2i 2811 . . . . 5 ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
38 finnum 9060 . . . . . . 7 (1𝑜 ∈ Fin → 1𝑜 ∈ dom card)
3913, 38ax-mp 5 . . . . . 6 1𝑜 ∈ dom card
40 carden2 9099 . . . . . 6 ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
4126, 39, 40sylancl 581 . . . . 5 (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
4237, 41syl5bbr 277 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) = 1𝑜𝐴 ≈ 1𝑜))
4334, 42orbi12d 943 . . 3 (𝐴 ∈ Fin → (((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜) ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)))
4422, 32, 433bitr3d 301 . 2 (𝐴 ∈ Fin → (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)))
458, 16, 44pm5.21nii 370 1 (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wo 874   = wceq 1653  wcel 2157  cin 3768  c0 4115  {cpr 4370   class class class wbr 4843  dom cdm 5312  Oncon0 5941  cfv 6101  ωcom 7299  1𝑜c1o 7792  2𝑜c2o 7793  cen 8192  cdom 8193  csdm 8194  Fincfn 8195  cardccrd 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7300  df-1o 7799  df-2o 7800  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051
This theorem is referenced by:  fin56  9503  en2top  21118
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