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Mirrors > Home > MPE Home > Th. List > fincssdom | Structured version Visualization version GIF version |
Description: In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
Ref | Expression |
---|---|
fincssdom | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1188 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐴 ∈ Fin) | |
2 | simpr 483 | . . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵) | |
3 | simpl3 1190 | . . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | |
4 | orel1 886 | . . . . . . . 8 ⊢ (¬ 𝐴 ⊆ 𝐵 → ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)) | |
5 | 2, 3, 4 | sylc 65 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ 𝐴) |
6 | dfpss3 4078 | . . . . . . 7 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵)) | |
7 | 5, 2, 6 | sylanbrc 581 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ⊊ 𝐴) |
8 | php3 9235 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | |
9 | 1, 7, 8 | syl2anc 582 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ≺ 𝐴) |
10 | 9 | ex 411 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (¬ 𝐴 ⊆ 𝐵 → 𝐵 ≺ 𝐴)) |
11 | domnsym 9122 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
12 | 11 | con2i 139 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐴 ≼ 𝐵) |
13 | 10, 12 | syl6 35 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 ≼ 𝐵)) |
14 | 13 | con4d 115 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ≼ 𝐵 → 𝐴 ⊆ 𝐵)) |
15 | ssdomg 9019 | . . 3 ⊢ (𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | |
16 | 15 | 3ad2ant2 1131 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
17 | 14, 16 | impbid 211 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 ∈ wcel 2098 ⊆ wss 3939 ⊊ wpss 3940 class class class wbr 5143 ≼ cdom 8960 ≺ csdm 8961 Fincfn 8962 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7869 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 |
This theorem is referenced by: fin1a2lem11 10433 |
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