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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 1057 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐴 ∈ Fin) | |
2 | simpr 476 | . . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵) | |
3 | simpl3 1059 | . . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | |
4 | orel1 396 | . . . . . . . 8 ⊢ (¬ 𝐴 ⊆ 𝐵 → ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)) | |
5 | 2, 3, 4 | sylc 63 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ 𝐴) |
6 | dfpss3 3655 | . . . . . . 7 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵)) | |
7 | 5, 2, 6 | sylanbrc 695 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ⊊ 𝐴) |
8 | php3 8031 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | |
9 | 1, 7, 8 | syl2anc 691 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ≺ 𝐴) |
10 | 9 | ex 449 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (¬ 𝐴 ⊆ 𝐵 → 𝐵 ≺ 𝐴)) |
11 | domnsym 7971 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
12 | 11 | con2i 133 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐴 ≼ 𝐵) |
13 | 10, 12 | syl6 34 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 ≼ 𝐵)) |
14 | 13 | con4d 113 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ≼ 𝐵 → 𝐴 ⊆ 𝐵)) |
15 | ssdomg 7887 | . . 3 ⊢ (𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | |
16 | 15 | 3ad2ant2 1076 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
17 | 14, 16 | impbid 201 | 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 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ⊆ wss 3540 ⊊ wpss 3541 class class class wbr 4583 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 |
This theorem is referenced by: fin1a2lem11 9115 |
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