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Theorem nnarcl 7936
Description: Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.)
Assertion
Ref Expression
nnarcl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))

Proof of Theorem nnarcl
StepHypRef Expression
1 oaword1 7872 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝐵))
2 eloni 5951 . . . . . . 7 (𝐴 ∈ On → Ord 𝐴)
3 ordom 7308 . . . . . . 7 Ord ω
4 ordtr2 5985 . . . . . . 7 ((Ord 𝐴 ∧ Ord ω) → ((𝐴 ⊆ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝐴 ∈ ω))
52, 3, 4sylancl 581 . . . . . 6 (𝐴 ∈ On → ((𝐴 ⊆ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝐴 ∈ ω))
65expd 405 . . . . 5 (𝐴 ∈ On → (𝐴 ⊆ (𝐴 +𝑜 𝐵) → ((𝐴 +𝑜 𝐵) ∈ ω → 𝐴 ∈ ω)))
76adantr 473 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ (𝐴 +𝑜 𝐵) → ((𝐴 +𝑜 𝐵) ∈ ω → 𝐴 ∈ ω)))
81, 7mpd 15 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) ∈ ω → 𝐴 ∈ ω))
9 oaword2 7873 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
109ancoms 451 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
11 eloni 5951 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
12 ordtr2 5985 . . . . . . 7 ((Ord 𝐵 ∧ Ord ω) → ((𝐵 ⊆ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝐵 ∈ ω))
1311, 3, 12sylancl 581 . . . . . 6 (𝐵 ∈ On → ((𝐵 ⊆ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝐵 ∈ ω))
1413expd 405 . . . . 5 (𝐵 ∈ On → (𝐵 ⊆ (𝐴 +𝑜 𝐵) → ((𝐴 +𝑜 𝐵) ∈ ω → 𝐵 ∈ ω)))
1514adantl 474 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ (𝐴 +𝑜 𝐵) → ((𝐴 +𝑜 𝐵) ∈ ω → 𝐵 ∈ ω)))
1610, 15mpd 15 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) ∈ ω → 𝐵 ∈ ω))
178, 16jcad 509 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) ∈ ω → (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))
18 nnacl 7931 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)
1917, 18impbid1 217 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wss 3769  Ord word 5940  Oncon0 5941  (class class class)co 6878  ωcom 7299   +𝑜 coa 7796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-oadd 7803
This theorem is referenced by:  finxpreclem4  33729
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