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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 9101 | . 2 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
2 | finnum 9945 | . . . . . 6 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ dom card) |
4 | finnum 9945 | . . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card) | |
5 | domtri2 9986 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) | |
6 | 3, 4, 5 | syl2an 595 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
7 | 6 | biimprd 247 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → (¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵)) |
8 | isinffi 9989 | . . . . . . 7 ⊢ ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑎 𝑎:𝐴–1-1→𝐵) | |
9 | 8 | ancoms 458 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑎 𝑎:𝐴–1-1→𝐵) |
10 | 9 | adantlr 712 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑎 𝑎:𝐴–1-1→𝐵) |
11 | brdomg 8954 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ≼ 𝐵 ↔ ∃𝑎 𝑎:𝐴–1-1→𝐵)) | |
12 | 11 | ad2antlr 724 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ∃𝑎 𝑎:𝐴–1-1→𝐵)) |
13 | 10, 12 | mpbird 257 | . . . 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 810 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → (¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵)) |
16 | 1, 15 | impbid2 225 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 class class class wbr 5141 dom cdm 5669 –1-1→wf1 6534 ≼ cdom 8939 ≺ csdm 8940 Fincfn 8941 cardccrd 9932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 |
This theorem is referenced by: fidomtri2 9991 fin56 10390 hauspwdom 23360 harinf 42351 safesnsupfidom1o 42744 |
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