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Theorem sdom2en01 9776
 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 8763 . . . . 5 ω = (On ∩ Fin)
2 inss2 4137 . . . . 5 (On ∩ Fin) ⊆ Fin
31, 2eqsstri 3929 . . . 4 ω ⊆ Fin
4 2onn 8283 . . . 4 2o ∈ ω
53, 4sselii 3892 . . 3 2o ∈ Fin
6 sdomdom 8569 . . 3 (𝐴 ≺ 2o𝐴 ≼ 2o)
7 domfi 8790 . . 3 ((2o ∈ Fin ∧ 𝐴 ≼ 2o) → 𝐴 ∈ Fin)
85, 6, 7sylancr 590 . 2 (𝐴 ≺ 2o𝐴 ∈ Fin)
9 id 22 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
10 0fin 8754 . . . 4 ∅ ∈ Fin
119, 10eqeltrdi 2861 . . 3 (𝐴 = ∅ → 𝐴 ∈ Fin)
12 1onn 8282 . . . . 5 1o ∈ ω
133, 12sselii 3892 . . . 4 1o ∈ Fin
14 enfi 8786 . . . 4 (𝐴 ≈ 1o → (𝐴 ∈ Fin ↔ 1o ∈ Fin))
1513, 14mpbiri 261 . . 3 (𝐴 ≈ 1o𝐴 ∈ Fin)
1611, 15jaoi 854 . 2 ((𝐴 = ∅ ∨ 𝐴 ≈ 1o) → 𝐴 ∈ Fin)
17 df2o3 8134 . . . . . 6 2o = {∅, 1o}
1817eleq2i 2844 . . . . 5 ((card‘𝐴) ∈ 2o ↔ (card‘𝐴) ∈ {∅, 1o})
19 fvex 6677 . . . . . 6 (card‘𝐴) ∈ V
2019elpr 4549 . . . . 5 ((card‘𝐴) ∈ {∅, 1o} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
2118, 20bitri 278 . . . 4 ((card‘𝐴) ∈ 2o ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
2221a1i 11 . . 3 (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o)))
23 cardnn 9439 . . . . . 6 (2o ∈ ω → (card‘2o) = 2o)
244, 23ax-mp 5 . . . . 5 (card‘2o) = 2o
2524eleq2i 2844 . . . 4 ((card‘𝐴) ∈ (card‘2o) ↔ (card‘𝐴) ∈ 2o)
26 finnum 9424 . . . . 5 (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27 2on 8128 . . . . . 6 2o ∈ On
28 onenon 9425 . . . . . 6 (2o ∈ On → 2o ∈ dom card)
2927, 28ax-mp 5 . . . . 5 2o ∈ dom card
30 cardsdom2 9464 . . . . 5 ((𝐴 ∈ dom card ∧ 2o ∈ dom card) → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
3126, 29, 30sylancl 589 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
3225, 31bitr3id 288 . . 3 (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o𝐴 ≺ 2o))
33 cardnueq0 9440 . . . . 5 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
3426, 33syl 17 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35 cardnn 9439 . . . . . . 7 (1o ∈ ω → (card‘1o) = 1o)
3612, 35ax-mp 5 . . . . . 6 (card‘1o) = 1o
3736eqeq2i 2772 . . . . 5 ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
38 finnum 9424 . . . . . . 7 (1o ∈ Fin → 1o ∈ dom card)
3913, 38ax-mp 5 . . . . . 6 1o ∈ dom card
40 carden2 9463 . . . . . 6 ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
4126, 39, 40sylancl 589 . . . . 5 (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
4237, 41bitr3id 288 . . . 4 (𝐴 ∈ Fin → ((card‘𝐴) = 1o𝐴 ≈ 1o))
4334, 42orbi12d 916 . . 3 (𝐴 ∈ Fin → (((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o) ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
4422, 32, 433bitr3d 312 . 2 (𝐴 ∈ Fin → (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
458, 16, 44pm5.21nii 383 1 (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 844   = wceq 1539   ∈ wcel 2112   ∩ cin 3860  ∅c0 4228  {cpr 4528   class class class wbr 5037  dom cdm 5529  Oncon0 6175  ‘cfv 6341  ωcom 7586  1oc1o 8112  2oc2o 8113   ≈ cen 8538   ≼ cdom 8539   ≺ csdm 8540  Fincfn 8541  cardccrd 9411 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-om 7587  df-1o 8119  df-2o 8120  df-er 8306  df-en 8542  df-dom 8543  df-sdom 8544  df-fin 8545  df-card 9415 This theorem is referenced by:  fin56  9867  en2top  21700
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