Theorem nnawordex 6775

Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnawordex	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnawordex

Step	Hyp	Ref	Expression
1		nntri3or 6739	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
2	1	3adant3 1044	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
3		nnaordex 6774	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
4		simpr 110	. . . . . . . 8 ⊢ ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → (𝐴 +o 𝑥) = 𝐵)
5	4	reximi 2641	. . . . . . 7 ⊢ (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
6	3, 5	biimtrdi 163	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
7	6	3adant3 1044	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
8		nna0 6720	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
9	8	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 +o ∅) = 𝐴)
10		eqeq2 2244	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝐴 +o ∅) = 𝐴 ↔ (𝐴 +o ∅) = 𝐵))
11	9, 10	syl5ibcom 155	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 = 𝐵 → (𝐴 +o ∅) = 𝐵))
12		peano1 4721	. . . . . . 7 ⊢ ∅ ∈ ω
13		oveq2 6066	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
14	13	eqeq1d 2243	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o ∅) = 𝐵))
15	14	rspcev 2923	. . . . . . 7 ⊢ ((∅ ∈ ω ∧ (𝐴 +o ∅) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
16	12, 15	mpan 424	. . . . . 6 ⊢ ((𝐴 +o ∅) = 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
17	11, 16	syl6 33	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐴 = 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
18		nntri1 6742	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
19	18	biimp3a 1382	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → ¬ 𝐵 ∈ 𝐴)
20	19	pm2.21d 624	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ 𝐴 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
21	7, 17, 20	3jaod 1341	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
22	2, 21	mpd 13	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
23	22	3expia 1232	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
24		nnaword1 6759	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
25		sseq2 3266	. . . . 5 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ 𝐴 ⊆ 𝐵))
26	24, 25	syl5ibcom 155	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
27	26	rexlimdva 2662	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
28	27	adantr 276	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
29	23, 28	impbid 129	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 1004   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∃wrex 2523   ⊆ wss 3214  ∅c0 3512  ωcom 4717  (class class class)co 6058   +_o coa 6657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664

This theorem is referenced by:  prarloclemn  7830