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Mirrors > Home > MPE Home > Th. List > nn2ge | Structured version Visualization version GIF version |
Description: There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
nn2ge | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 12243 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ) |
3 | nnre 12243 | . . 3 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
4 | 3 | adantl 481 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ) |
5 | leid 11334 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ 𝐵) | |
6 | 5 | anim1ci 615 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)) |
7 | 3, 6 | sylan 579 | . . . 4 ⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)) |
8 | breq2 5146 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ≤ 𝑥 ↔ 𝐴 ≤ 𝐵)) | |
9 | breq2 5146 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝐵)) | |
10 | 8, 9 | anbi12d 631 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥) ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))) |
11 | 10 | rspcev 3608 | . . . 4 ⊢ ((𝐵 ∈ ℕ ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) |
12 | 7, 11 | syldan 590 | . . 3 ⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) |
13 | 12 | adantll 713 | . 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝐴 ≤ 𝐵) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) |
14 | leid 11334 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | |
15 | 14 | anim1i 614 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ≤ 𝐴) → (𝐴 ≤ 𝐴 ∧ 𝐵 ≤ 𝐴)) |
16 | 1, 15 | sylan 579 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ≤ 𝐴) → (𝐴 ≤ 𝐴 ∧ 𝐵 ≤ 𝐴)) |
17 | breq2 5146 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ≤ 𝑥 ↔ 𝐴 ≤ 𝐴)) | |
18 | breq2 5146 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝐴)) | |
19 | 17, 18 | anbi12d 631 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥) ↔ (𝐴 ≤ 𝐴 ∧ 𝐵 ≤ 𝐴))) |
20 | 19 | rspcev 3608 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ (𝐴 ≤ 𝐴 ∧ 𝐵 ≤ 𝐴)) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) |
21 | 16, 20 | syldan 590 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ≤ 𝐴) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) |
22 | 21 | adantlr 714 | . 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝐵 ≤ 𝐴) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) |
23 | 2, 4, 13, 22 | lecasei 11344 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3066 class class class wbr 5142 ℝcr 11131 ≤ cle 11273 ℕcn 12236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-i2m1 11200 ax-1ne0 11201 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-nn 12237 |
This theorem is referenced by: (None) |
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