![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
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 12946 | . . 3 ⊢ ℚ ∈ V | |
2 | 1 | pwex 5371 | . 2 ⊢ 𝒫 ℚ ∈ V |
3 | ssrab2 4072 | . . . . 5 ⊢ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ⊆ ℚ | |
4 | 1 | elpw2 5338 | . . . . 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 11297 | . . . . . 6 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 ≠ 𝑏 ↔ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎))) | |
8 | rpnnen3lem 42330 | . . . . . . . 8 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) | |
9 | rpnnen3lem 42330 | . . . . . . . . . 10 ⊢ (((𝑏 ∈ ℝ ∧ 𝑎 ∈ ℝ) ∧ 𝑏 < 𝑎) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎}) | |
10 | 9 | ancom1s 650 | . . . . . . . . 9 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑏 < 𝑎) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎}) |
11 | 10 | necomd 2990 | . . . . . . . 8 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑏 < 𝑎) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
12 | 8, 11 | jaodan 954 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (𝑎 < 𝑏 ∨ 𝑏 < 𝑎)) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
13 | 12 | ex 412 | . . . . . 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 2958 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → ({𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} = {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} → 𝑎 = 𝑏)) |
16 | breq2 5145 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝑐 < 𝑎 ↔ 𝑐 < 𝑏)) | |
17 | 16 | rabbidv 3434 | . . . 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 8990 | . 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 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 {crab 3426 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 class class class wbr 5141 ≼ cdom 8936 ℝcr 11108 < clt 11249 ℚcq 12933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |