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Mirrors > Home > MPE Home > Th. List > omord | Structured version Visualization version GIF version |
Description: Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. Theorem 3.16 of [Schloeder] p. 9. (Contributed by NM, 14-Dec-2004.) |
Ref | Expression |
---|---|
omord | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omord2 8604 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) | |
2 | 1 | ex 412 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))) |
3 | 2 | pm5.32rd 578 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∧ ∅ ∈ 𝐶))) |
4 | simpl 482 | . . 3 ⊢ (((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)) | |
5 | ne0i 4347 | . . . . . . . 8 ⊢ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (𝐶 ·o 𝐵) ≠ ∅) | |
6 | om0r 8576 | . . . . . . . . . 10 ⊢ (𝐵 ∈ On → (∅ ·o 𝐵) = ∅) | |
7 | oveq1 7438 | . . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵)) | |
8 | 7 | eqeq1d 2737 | . . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((𝐶 ·o 𝐵) = ∅ ↔ (∅ ·o 𝐵) = ∅)) |
9 | 6, 8 | syl5ibrcom 247 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝐶 = ∅ → (𝐶 ·o 𝐵) = ∅)) |
10 | 9 | necon3d 2959 | . . . . . . . 8 ⊢ (𝐵 ∈ On → ((𝐶 ·o 𝐵) ≠ ∅ → 𝐶 ≠ ∅)) |
11 | 5, 10 | syl5 34 | . . . . . . 7 ⊢ (𝐵 ∈ On → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐶 ≠ ∅)) |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐶 ≠ ∅)) |
13 | on0eln0 6442 | . . . . . . 7 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅)) | |
14 | 13 | adantl 481 | . . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅)) |
15 | 12, 14 | sylibrd 259 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ∅ ∈ 𝐶)) |
16 | 15 | 3adant1 1129 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ∅ ∈ 𝐶)) |
17 | 16 | ancld 550 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∧ ∅ ∈ 𝐶))) |
18 | 4, 17 | impbid2 226 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) |
19 | 3, 18 | bitrd 279 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 Oncon0 6386 (class class class)co 7431 ·o comu 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-oadd 8509 df-omul 8510 |
This theorem is referenced by: omlimcl 8615 oneo 8618 omord2lim 43290 omord2com 43292 |
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