Theorem bndndx 9307

Description: A bounded real sequence 𝐴(𝑘) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)

Assertion

Ref	Expression
bndndx	⊢ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑘

Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem bndndx

Step	Hyp	Ref	Expression
1		arch 9305	. . . 4 ⊢ (𝑥 ∈ ℝ → ∃𝑘 ∈ ℕ 𝑥 < 𝑘)
2		nnre 9056	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
3		lelttr 8174	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝑘) → 𝐴 < 𝑘))
4		ltle 8173	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐴 < 𝑘 → 𝐴 ≤ 𝑘))
5	4	3adant2 1019	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐴 < 𝑘 → 𝐴 ≤ 𝑘))
6	3, 5	syld 45	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝑘) → 𝐴 ≤ 𝑘))
7	6	exp5o 1229	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝑥 ∈ ℝ → (𝑘 ∈ ℝ → (𝐴 ≤ 𝑥 → (𝑥 < 𝑘 → 𝐴 ≤ 𝑘)))))
8	7	com3l 81	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ ℝ → (𝐴 ∈ ℝ → (𝐴 ≤ 𝑥 → (𝑥 < 𝑘 → 𝐴 ≤ 𝑘)))))
9	8	imp4b 350	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → (𝑥 < 𝑘 → 𝐴 ≤ 𝑘)))
10	9	com23 78	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑥 < 𝑘 → ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘)))
11	2, 10	sylan2 286	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑥 < 𝑘 → ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘)))
12	11	reximdva 2609	. . . 4 ⊢ (𝑥 ∈ ℝ → (∃𝑘 ∈ ℕ 𝑥 < 𝑘 → ∃𝑘 ∈ ℕ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘)))
13	1, 12	mpd 13	. . 3 ⊢ (𝑥 ∈ ℝ → ∃𝑘 ∈ ℕ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘))
14		r19.35-1 2657	. . 3 ⊢ (∃𝑘 ∈ ℕ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘) → (∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘))
15	13, 14	syl 14	. 2 ⊢ (𝑥 ∈ ℝ → (∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘))
16	15	rexlimiv 2618	1 ⊢ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486   class class class wbr 4048  ℝcr 7937   < clt 8120   ≤ cle 8121  ℕcn 9049

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1re 8032  ax-addrcl 8035  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-arch 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-xp 4686  df-cnv 4688  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-inn 9050

This theorem is referenced by: (None)