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Mirrors > Home > MPE Home > Th. List > nnarcl | Structured version Visualization version GIF version |
Description: Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.) |
Ref | Expression |
---|---|
nnarcl | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oaword1 8383 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵)) | |
2 | eloni 6276 | . . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordom 7722 | . . . . . . 7 ⊢ Ord ω | |
4 | ordtr2 6310 | . . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → ((𝐴 ⊆ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝐴 ∈ ω)) | |
5 | 2, 3, 4 | sylancl 586 | . . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ⊆ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝐴 ∈ ω)) |
6 | 5 | expd 416 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (𝐴 +o 𝐵) → ((𝐴 +o 𝐵) ∈ ω → 𝐴 ∈ ω))) |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ (𝐴 +o 𝐵) → ((𝐴 +o 𝐵) ∈ ω → 𝐴 ∈ ω))) |
8 | 1, 7 | mpd 15 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω → 𝐴 ∈ ω)) |
9 | oaword2 8384 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵)) | |
10 | 9 | ancoms 459 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵)) |
11 | eloni 6276 | . . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
12 | ordtr2 6310 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord ω) → ((𝐵 ⊆ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝐵 ∈ ω)) | |
13 | 11, 3, 12 | sylancl 586 | . . . . . 6 ⊢ (𝐵 ∈ On → ((𝐵 ⊆ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝐵 ∈ ω)) |
14 | 13 | expd 416 | . . . . 5 ⊢ (𝐵 ∈ On → (𝐵 ⊆ (𝐴 +o 𝐵) → ((𝐴 +o 𝐵) ∈ ω → 𝐵 ∈ ω))) |
15 | 14 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ (𝐴 +o 𝐵) → ((𝐴 +o 𝐵) ∈ ω → 𝐵 ∈ ω))) |
16 | 10, 15 | mpd 15 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω → 𝐵 ∈ ω)) |
17 | 8, 16 | jcad 513 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))) |
18 | nnacl 8442 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω) | |
19 | 17, 18 | impbid1 224 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ⊆ wss 3887 Ord word 6265 Oncon0 6266 (class class class)co 7275 ωcom 7712 +o coa 8294 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-oadd 8301 |
This theorem is referenced by: finxpreclem4 35565 |
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