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Mirrors > Home > MPE Home > Th. List > fidomtri | Structured version Visualization version GIF version |
Description: Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
fidomtri | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnsym 8240 | . 2 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
2 | finnum 8972 | . . . . . 6 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
3 | 2 | adantr 466 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ dom card) |
4 | finnum 8972 | . . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card) | |
5 | domtri2 9013 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) | |
6 | 3, 4, 5 | syl2an 583 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
7 | 6 | biimprd 238 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → (¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵)) |
8 | isinffi 9016 | . . . . . . 7 ⊢ ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑎 𝑎:𝐴–1-1→𝐵) | |
9 | 8 | ancoms 446 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑎 𝑎:𝐴–1-1→𝐵) |
10 | 9 | adantlr 694 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑎 𝑎:𝐴–1-1→𝐵) |
11 | brdomg 8117 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ≼ 𝐵 ↔ ∃𝑎 𝑎:𝐴–1-1→𝐵)) | |
12 | 11 | ad2antlr 706 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ∃𝑎 𝑎:𝐴–1-1→𝐵)) |
13 | 10, 12 | mpbird 247 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → 𝐴 ≼ 𝐵) |
14 | 13 | a1d 25 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵)) |
15 | 7, 14 | pm2.61dan 814 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → (¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵)) |
16 | 1, 15 | impbid2 216 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∃wex 1852 ∈ wcel 2145 class class class wbr 4786 dom cdm 5249 –1-1→wf1 6026 ≼ cdom 8105 ≺ csdm 8106 Fincfn 8107 cardccrd 8959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-om 7211 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-card 8963 |
This theorem is referenced by: fidomtri2 9018 fin56 9415 hauspwdom 21518 harinf 38120 |
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