Theorem xnn0lenn0nn0 9987

Description: An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.)

Assertion

Ref	Expression
xnn0lenn0nn0	⊢ ((𝑀 ∈ ℕ0* ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ0)

Proof of Theorem xnn0lenn0nn0

Step	Hyp	Ref	Expression
1		elxnn0 9360	. . 3 ⊢ (𝑀 ∈ ℕ0* ↔ (𝑀 ∈ ℕ0 ∨ 𝑀 = +∞))
2		2a1 25	. . . 4 ⊢ (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0)))
3		breq1 4047	. . . . . . 7 ⊢ (𝑀 = +∞ → (𝑀 ≤ 𝑁 ↔ +∞ ≤ 𝑁))
4	3	adantr 276	. . . . . 6 ⊢ ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ +∞ ≤ 𝑁))
5		nn0re 9304	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
6	5	rexrd 8122	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ*)
7		xgepnf 9938	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ* → (+∞ ≤ 𝑁 ↔ 𝑁 = +∞))
8	6, 7	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (+∞ ≤ 𝑁 ↔ 𝑁 = +∞))
9		pnfnre 8114	. . . . . . . . 9 ⊢ +∞ ∉ ℝ
10		eleq1 2268	. . . . . . . . . . 11 ⊢ (𝑁 = +∞ → (𝑁 ∈ ℕ0 ↔ +∞ ∈ ℕ0))
11		nn0re 9304	. . . . . . . . . . . 12 ⊢ (+∞ ∈ ℕ0 → +∞ ∈ ℝ)
12		elnelall 2483	. . . . . . . . . . . 12 ⊢ (+∞ ∈ ℝ → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0))
13	11, 12	syl 14	. . . . . . . . . . 11 ⊢ (+∞ ∈ ℕ0 → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0))
14	10, 13	biimtrdi 163	. . . . . . . . . 10 ⊢ (𝑁 = +∞ → (𝑁 ∈ ℕ0 → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0)))
15	14	com13 80	. . . . . . . . 9 ⊢ (+∞ ∉ ℝ → (𝑁 ∈ ℕ0 → (𝑁 = +∞ → 𝑀 ∈ ℕ0)))
16	9, 15	ax-mp 5	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 = +∞ → 𝑀 ∈ ℕ0))
17	8, 16	sylbid 150	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (+∞ ≤ 𝑁 → 𝑀 ∈ ℕ0))
18	17	adantl 277	. . . . . 6 ⊢ ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (+∞ ≤ 𝑁 → 𝑀 ∈ ℕ0))
19	4, 18	sylbid 150	. . . . 5 ⊢ ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0))
20	19	ex 115	. . . 4 ⊢ (𝑀 = +∞ → (𝑁 ∈ ℕ0 → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0)))
21	2, 20	jaoi 718	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∨ 𝑀 = +∞) → (𝑁 ∈ ℕ0 → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0)))
22	1, 21	sylbi 121	. 2 ⊢ (𝑀 ∈ ℕ0* → (𝑁 ∈ ℕ0 → (𝑀 ≤ 𝑁 → 𝑀 ∈ ℕ0)))
23	22	3imp 1196	1 ⊢ ((𝑀 ∈ ℕ0* ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ0)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2176   ∉ wnel 2471   class class class wbr 4044  ℝcr 7924  +∞cpnf 8104  ℝ^*cxr 8106   ≤ cle 8108  ℕ₀cn0 9295  ℕ₀^*cxnn0 9358

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022  ax-rnegex 8034  ax-pre-ltirr 8037

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-inn 9037  df-n0 9296  df-xnn0 9359

This theorem is referenced by:  xnn0le2is012  9988