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Mirrors > Home > MPE Home > Th. List > omword1 | Structured version Visualization version GIF version |
Description: An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.) |
Ref | Expression |
---|---|
omword1 | ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 5733 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
2 | ordgt0ge1 7577 | . . . . 5 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1𝑜 ⊆ 𝐵)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1𝑜 ⊆ 𝐵)) |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1𝑜 ⊆ 𝐵)) |
5 | 1on 7567 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
6 | omwordi 7651 | . . . . . 6 ⊢ ((1𝑜 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜 ⊆ 𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵))) | |
7 | 5, 6 | mp3an1 1411 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜 ⊆ 𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵))) |
8 | 7 | ancoms 469 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 ⊆ 𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵))) |
9 | om1 7622 | . . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 1𝑜) = 𝐴) | |
10 | 9 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 1𝑜) = 𝐴) |
11 | 10 | sseq1d 3632 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵) ↔ 𝐴 ⊆ (𝐴 ·𝑜 𝐵))) |
12 | 8, 11 | sylibd 229 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))) |
13 | 4, 12 | sylbid 230 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))) |
14 | 13 | imp 445 | 1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∅c0 3915 Ord word 5722 Oncon0 5723 (class class class)co 6650 1𝑜c1o 7553 ·𝑜 comu 7558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 |
This theorem is referenced by: om00 7655 cantnflem3 8588 cantnflem4 8589 cnfcomlem 8596 |
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