Theorem nn2get 4438

Description: There exists a natural number greater than or equal to any two others.

Assertion

Ref	Expression
nn2get	⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x))

Distinct variable group(s):   x,A   x,B

Proof of Theorem nn2get

Step	Hyp	Ref	Expression
1		letrit 4347	. . 3 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A ≤ B ∨ B ≤ A))
2		nnret 4427	. . 3 ⊢ (A ∈ ℕ → A ∈ ℝ)
3		nnret 4427	. . 3 ⊢ (B ∈ ℕ → B ∈ ℝ)
4	1, 2, 3	syl2an 349	. 2 ⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (A ≤ B ∨ B ≤ A))
5		leidt 4293	. . . . . . . . 9 ⊢ (B ∈ ℝ → B ≤ B)
6	5	biantrud 545	. . . . . . . 8 ⊢ (B ∈ ℝ → (A ≤ B ↔ (A ≤ B ∧ B ≤ B)))
7	6	biimpd 135	. . . . . . 7 ⊢ (B ∈ ℝ → (A ≤ B → (A ≤ B ∧ B ≤ B)))
8	3, 7	syl 12	. . . . . 6 ⊢ (B ∈ ℕ → (A ≤ B → (A ≤ B ∧ B ≤ B)))
9	8	anc2li 250	. . . . 5 ⊢ (B ∈ ℕ → (A ≤ B → (B ∈ ℕ ∧ (A ≤ B ∧ B ≤ B))))
10		breq2 2066	. . . . . . 7 ⊢ (x = B → (A ≤ x ↔ A ≤ B))
11		breq2 2066	. . . . . . 7 ⊢ (x = B → (B ≤ x ↔ B ≤ B))
12	10, 11	anbi12d 476	. . . . . 6 ⊢ (x = B → ((A ≤ x ∧ B ≤ x) ↔ (A ≤ B ∧ B ≤ B)))
13	12	rcla4ev 1403	. . . . 5 ⊢ ((B ∈ ℕ ∧ (A ≤ B ∧ B ≤ B)) → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x))
14	9, 13	syl6 23	. . . 4 ⊢ (B ∈ ℕ → (A ≤ B → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x)))
15	14	adantl 305	. . 3 ⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (A ≤ B → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x)))
16		leidt 4293	. . . . . . . . 9 ⊢ (A ∈ ℝ → A ≤ A)
17	16	biantrurd 546	. . . . . . . 8 ⊢ (A ∈ ℝ → (B ≤ A ↔ (A ≤ A ∧ B ≤ A)))
18	17	biimpd 135	. . . . . . 7 ⊢ (A ∈ ℝ → (B ≤ A → (A ≤ A ∧ B ≤ A)))
19	2, 18	syl 12	. . . . . 6 ⊢ (A ∈ ℕ → (B ≤ A → (A ≤ A ∧ B ≤ A)))
20	19	anc2li 250	. . . . 5 ⊢ (A ∈ ℕ → (B ≤ A → (A ∈ ℕ ∧ (A ≤ A ∧ B ≤ A))))
21		breq2 2066	. . . . . . 7 ⊢ (x = A → (A ≤ x ↔ A ≤ A))
22		breq2 2066	. . . . . . 7 ⊢ (x = A → (B ≤ x ↔ B ≤ A))
23	21, 22	anbi12d 476	. . . . . 6 ⊢ (x = A → ((A ≤ x ∧ B ≤ x) ↔ (A ≤ A ∧ B ≤ A)))
24	23	rcla4ev 1403	. . . . 5 ⊢ ((A ∈ ℕ ∧ (A ≤ A ∧ B ≤ A)) → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x))
25	20, 24	syl6 23	. . . 4 ⊢ (A ∈ ℕ → (B ≤ A → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x)))
26	25	adantr 306	. . 3 ⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (B ≤ A → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x)))
27	15, 26	jaod 329	. 2 ⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → ((A ≤ B ∨ B ≤ A) → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x)))
28	4, 27	mpd 46	1 ⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → ∃x ∈ ℕ (A ≤ x ∧ B ≤ x))

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   class class class wbr 2054  ℝcr 4027   ≤ cle 4092  ℕcn 4093

This theorem is referenced by:  climunii 4883  hlimunii 5143

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962  df-ltr 3964  df-0r 3965  df-1r 3966  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-lt 4041  df-le 4277  df-n 4423

metamath.org