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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rpnnen3 | Structured version Visualization version GIF version |
Description: Dedekind cut injection of ℝ into 𝒫 ℚ. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Ref | Expression |
---|---|
rpnnen3 | ⊢ ℝ ≼ 𝒫 ℚ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qex 12997 | . . 3 ⊢ ℚ ∈ V | |
2 | 1 | pwex 5384 | . 2 ⊢ 𝒫 ℚ ∈ V |
3 | ssrab2 4076 | . . . . 5 ⊢ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ⊆ ℚ | |
4 | 1 | elpw2 5352 | . . . . 5 ⊢ ({𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ∈ 𝒫 ℚ ↔ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ⊆ ℚ) |
5 | 3, 4 | mpbir 230 | . . . 4 ⊢ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ∈ 𝒫 ℚ |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑎 ∈ ℝ → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ∈ 𝒫 ℚ) |
7 | lttri2 11346 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎))) | |
8 | rpnnen3lem 42689 | . . . . . . . 8 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) | |
9 | rpnnen3lem 42689 | . . . . . . . . . 10 ⊢ (((𝑏 ∈ ℝ ∧ 𝑎 ∈ ℝ) ∧ 𝑏 < 𝑎) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎}) | |
10 | 9 | ancom1s 651 | . . . . . . . . 9 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑏 < 𝑎) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎}) |
11 | 10 | necomd 2986 | . . . . . . . 8 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑏 < 𝑎) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
12 | 8, 11 | jaodan 955 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
13 | 12 | ex 411 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((𝑎 < 𝑏 ∨ 𝑏 < 𝑎) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏})) |
14 | 7, 13 | sylbid 239 | . . . . 5 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏})) |
15 | 14 | necon4d 2954 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ({𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} = {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} → 𝑎 = 𝑏)) |
16 | breq2 5157 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝑐 < 𝑎 ↔ 𝑐 < 𝑏)) | |
17 | 16 | rabbidv 3427 | . . . 4 ⊢ (𝑎 = 𝑏 → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} = {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
18 | 15, 17 | impbid1 224 | . . 3 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ({𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} = {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ↔ 𝑎 = 𝑏)) |
19 | 6, 18 | dom2 9026 | . 2 ⊢ (𝒫 ℚ ∈ V → ℝ ≼ 𝒫 ℚ) |
20 | 2, 19 | ax-mp 5 | 1 ⊢ ℝ ≼ 𝒫 ℚ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 {crab 3419 Vcvv 3462 ⊆ wss 3947 𝒫 cpw 4607 class class class wbr 5153 ≼ cdom 8972 ℝcr 11157 < clt 11298 ℚcq 12984 |
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 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 |
This theorem is referenced by: (None) |
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