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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 8172 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵)) | |
2 | eloni 6196 | . . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordom 7583 | . . . . . . 7 ⊢ Ord ω | |
4 | ordtr2 6230 | . . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → ((𝐴 ⊆ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝐴 ∈ ω)) | |
5 | 2, 3, 4 | sylancl 588 | . . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ⊆ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝐴 ∈ ω)) |
6 | 5 | expd 418 | . . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (𝐴 +o 𝐵) → ((𝐴 +o 𝐵) ∈ ω → 𝐴 ∈ ω))) |
7 | 6 | adantr 483 | . . . 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 8173 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵)) | |
10 | 9 | ancoms 461 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 +o 𝐵)) |
11 | eloni 6196 | . . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
12 | ordtr2 6230 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord ω) → ((𝐵 ⊆ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝐵 ∈ ω)) | |
13 | 11, 3, 12 | sylancl 588 | . . . . . 6 ⊢ (𝐵 ∈ On → ((𝐵 ⊆ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝐵 ∈ ω)) |
14 | 13 | expd 418 | . . . . 5 ⊢ (𝐵 ∈ On → (𝐵 ⊆ (𝐴 +o 𝐵) → ((𝐴 +o 𝐵) ∈ ω → 𝐵 ∈ ω))) |
15 | 14 | adantl 484 | . . . 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 515 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))) |
18 | nnacl 8231 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω) | |
19 | 17, 18 | impbid1 227 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ⊆ wss 3936 Ord word 6185 Oncon0 6186 (class class class)co 7150 ωcom 7574 +o coa 8093 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 |
This theorem is referenced by: finxpreclem4 34669 |
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