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Mirrors > Home > MPE Home > Th. List > zextle | Structured version Visualization version GIF version |
Description: An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.) |
Ref | Expression |
---|---|
zextle | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑀 = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 12180 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
2 | 1 | leidd 11398 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ≤ 𝑀) |
3 | 2 | adantr 484 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑀 ≤ 𝑀) |
4 | breq1 5056 | . . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑘 ≤ 𝑀 ↔ 𝑀 ≤ 𝑀)) | |
5 | breq1 5056 | . . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (𝑘 ≤ 𝑁 ↔ 𝑀 ≤ 𝑁)) | |
6 | 4, 5 | bibi12d 349 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → ((𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁) ↔ (𝑀 ≤ 𝑀 ↔ 𝑀 ≤ 𝑁))) |
7 | 6 | rspcva 3535 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → (𝑀 ≤ 𝑀 ↔ 𝑀 ≤ 𝑁)) |
8 | 3, 7 | mpbid 235 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑀 ≤ 𝑁) |
9 | 8 | adantlr 715 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑀 ≤ 𝑁) |
10 | zre 12180 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
11 | 10 | leidd 11398 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ≤ 𝑁) |
12 | 11 | adantr 484 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑁 ≤ 𝑁) |
13 | breq1 5056 | . . . . . . . . 9 ⊢ (𝑘 = 𝑁 → (𝑘 ≤ 𝑀 ↔ 𝑁 ≤ 𝑀)) | |
14 | breq1 5056 | . . . . . . . . 9 ⊢ (𝑘 = 𝑁 → (𝑘 ≤ 𝑁 ↔ 𝑁 ≤ 𝑁)) | |
15 | 13, 14 | bibi12d 349 | . . . . . . . 8 ⊢ (𝑘 = 𝑁 → ((𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁) ↔ (𝑁 ≤ 𝑀 ↔ 𝑁 ≤ 𝑁))) |
16 | 15 | rspcva 3535 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → (𝑁 ≤ 𝑀 ↔ 𝑁 ≤ 𝑁)) |
17 | 12, 16 | mpbird 260 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑁 ≤ 𝑀) |
18 | 17 | adantll 714 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑁 ≤ 𝑀) |
19 | 9, 18 | jca 515 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) |
20 | 19 | ex 416 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁) → (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))) |
21 | letri3 10918 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 = 𝑁 ↔ (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))) | |
22 | 1, 10, 21 | syl2an 599 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 𝑁 ↔ (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))) |
23 | 20, 22 | sylibrd 262 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁) → 𝑀 = 𝑁)) |
24 | 23 | 3impia 1119 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑀 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 class class class wbr 5053 ℝcr 10728 ≤ cle 10868 ℤcz 12176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-neg 11065 df-z 12177 |
This theorem is referenced by: zextlt 12251 |
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