Theorem nnarcl 7854

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

Step	Hyp	Ref	Expression
1		oaword1 7790	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝐵))
2		eloni 5875	. . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordom 7225	. . . . . . 7 ⊢ Ord ω
4		ordtr2 5910	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord ω) → ((𝐴 ⊆ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝐴 ∈ ω))
5	2, 3, 4	sylancl 574	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝐴 ⊆ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝐴 ∈ ω))
6	5	expd 400	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ⊆ (𝐴 +𝑜 𝐵) → ((𝐴 +𝑜 𝐵) ∈ ω → 𝐴 ∈ ω)))
7	6	adantr 466	. . . 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 7791	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
10	9	ancoms 446	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
11		eloni 5875	. . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵)
12		ordtr2 5910	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord ω) → ((𝐵 ⊆ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝐵 ∈ ω))
13	11, 3, 12	sylancl 574	. . . . . 6 ⊢ (𝐵 ∈ On → ((𝐵 ⊆ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝐵 ∈ ω))
14	13	expd 400	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 ⊆ (𝐴 +𝑜 𝐵) → ((𝐴 +𝑜 𝐵) ∈ ω → 𝐵 ∈ ω)))
15	14	adantl 467	. . . 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 502	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) ∈ ω → (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))
18		nnacl 7849	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)
19	17, 18	impbid1 215	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∈ wcel 2145   ⊆ wss 3723  Ord word 5864  Oncon0 5865  (class class class)co 6796  ωcom 7216   +_𝑜 coa 7714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-oadd 7721

This theorem is referenced by:  finxpreclem4  33568