![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > rpnnen3lem | Structured version Visualization version GIF version |
Description: Lemma for rpnnen3 42354. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Ref | Expression |
---|---|
rpnnen3lem | ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qbtwnre 13184 | . . 3 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) → ∃𝑑 ∈ ℚ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) | |
2 | simp2 1134 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → 𝑑 ∈ ℚ) | |
3 | simp3r 1199 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → 𝑑 < 𝑏) | |
4 | breq1 5144 | . . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝑐 < 𝑏 ↔ 𝑑 < 𝑏)) | |
5 | 4 | elrab 3678 | . . . . . . 7 ⊢ (𝑑 ∈ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ↔ (𝑑 ∈ ℚ ∧ 𝑑 < 𝑏)) |
6 | 2, 3, 5 | sylanbrc 582 | . . . . . 6 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → 𝑑 ∈ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
7 | simp11 1200 | . . . . . . . . 9 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → 𝑎 ∈ ℝ) | |
8 | qre 12941 | . . . . . . . . . 10 ⊢ (𝑑 ∈ ℚ → 𝑑 ∈ ℝ) | |
9 | 8 | 3ad2ant2 1131 | . . . . . . . . 9 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → 𝑑 ∈ ℝ) |
10 | simp3l 1198 | . . . . . . . . 9 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → 𝑎 < 𝑑) | |
11 | 7, 9, 10 | ltnsymd 11367 | . . . . . . . 8 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → ¬ 𝑑 < 𝑎) |
12 | 11 | intnand 488 | . . . . . . 7 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → ¬ (𝑑 ∈ ℚ ∧ 𝑑 < 𝑎)) |
13 | breq1 5144 | . . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝑐 < 𝑎 ↔ 𝑑 < 𝑎)) | |
14 | 13 | elrab 3678 | . . . . . . 7 ⊢ (𝑑 ∈ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ↔ (𝑑 ∈ ℚ ∧ 𝑑 < 𝑎)) |
15 | 12, 14 | sylnibr 329 | . . . . . 6 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → ¬ 𝑑 ∈ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎}) |
16 | nelne1 3033 | . . . . . 6 ⊢ ((𝑑 ∈ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ∧ ¬ 𝑑 ∈ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎}) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎}) | |
17 | 6, 15, 16 | syl2anc 583 | . . . . 5 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎}) |
18 | 17 | necomd 2990 | . . . 4 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) ∧ 𝑑 ∈ ℚ ∧ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏)) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
19 | 18 | rexlimdv3a 3153 | . . 3 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) → (∃𝑑 ∈ ℚ (𝑎 < 𝑑 ∧ 𝑑 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏})) |
20 | 1, 19 | mpd 15 | . 2 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
21 | 20 | 3expa 1115 | 1 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2934 ∃wrex 3064 {crab 3426 class class class wbr 5141 ℝcr 11111 < clt 11252 ℚcq 12936 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 |
This theorem is referenced by: rpnnen3 42354 |
Copyright terms: Public domain | W3C validator |