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Mirrors > Home > MPE Home > Th. List > smoword | Structured version Visualization version GIF version |
Description: A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.) |
Ref | Expression |
---|---|
smoword | ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ⊆ 𝐷 ↔ (𝐹‘𝐶) ⊆ (𝐹‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smoord 8312 | . . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐷 ∈ 𝐶 ↔ (𝐹‘𝐷) ∈ (𝐹‘𝐶))) | |
2 | 1 | notbid 318 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐷 ∈ 𝐶 ↔ ¬ (𝐹‘𝐷) ∈ (𝐹‘𝐶))) |
3 | 2 | ancom2s 649 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ 𝐷 ∈ 𝐶 ↔ ¬ (𝐹‘𝐷) ∈ (𝐹‘𝐶))) |
4 | smodm2 8302 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴) | |
5 | simprl 770 | . . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ 𝐴) | |
6 | ordelord 6340 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶) | |
7 | 4, 5, 6 | syl2an2r 684 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → Ord 𝐶) |
8 | simprr 772 | . . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ 𝐴) | |
9 | ordelord 6340 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐷 ∈ 𝐴) → Ord 𝐷) | |
10 | 4, 8, 9 | syl2an2r 684 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → Ord 𝐷) |
11 | ordtri1 6351 | . . 3 ⊢ ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 ⊆ 𝐷 ↔ ¬ 𝐷 ∈ 𝐶)) | |
12 | 7, 10, 11 | syl2anc 585 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ⊆ 𝐷 ↔ ¬ 𝐷 ∈ 𝐶)) |
13 | simplr 768 | . . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → Smo 𝐹) | |
14 | smofvon2 8303 | . . . 4 ⊢ (Smo 𝐹 → (𝐹‘𝐶) ∈ On) | |
15 | eloni 6328 | . . . 4 ⊢ ((𝐹‘𝐶) ∈ On → Ord (𝐹‘𝐶)) | |
16 | 13, 14, 15 | 3syl 18 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → Ord (𝐹‘𝐶)) |
17 | smofvon2 8303 | . . . 4 ⊢ (Smo 𝐹 → (𝐹‘𝐷) ∈ On) | |
18 | eloni 6328 | . . . 4 ⊢ ((𝐹‘𝐷) ∈ On → Ord (𝐹‘𝐷)) | |
19 | 13, 17, 18 | 3syl 18 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → Ord (𝐹‘𝐷)) |
20 | ordtri1 6351 | . . 3 ⊢ ((Ord (𝐹‘𝐶) ∧ Ord (𝐹‘𝐷)) → ((𝐹‘𝐶) ⊆ (𝐹‘𝐷) ↔ ¬ (𝐹‘𝐷) ∈ (𝐹‘𝐶))) | |
21 | 16, 19, 20 | syl2anc 585 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) ⊆ (𝐹‘𝐷) ↔ ¬ (𝐹‘𝐷) ∈ (𝐹‘𝐶))) |
22 | 3, 12, 21 | 3bitr4d 311 | 1 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ⊆ 𝐷 ↔ (𝐹‘𝐶) ⊆ (𝐹‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3911 Ord word 6317 Oncon0 6318 Fn wfn 6492 ‘cfv 6497 Smo wsmo 8292 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-ord 6321 df-on 6322 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-smo 8293 |
This theorem is referenced by: cfcoflem 10213 coftr 10214 |
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