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Mirrors > Home > ILE Home > Th. List > lteupri | GIF version |
Description: The difference from ltexpri 6935 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.) |
Ref | Expression |
---|---|
lteupri | ⊢ (𝐴<P 𝐵 → ∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltexpri 6935 | . 2 ⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | |
2 | ltrelpr 6827 | . . . . 5 ⊢ <P ⊆ (P × P) | |
3 | 2 | brel 4438 | . . . 4 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
4 | 3 | simpld 110 | . . 3 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P) |
5 | eqtr3 2102 | . . . . . . . 8 ⊢ (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → (𝐴 +P 𝑥) = (𝐴 +P 𝑦)) | |
6 | addcanprg 6938 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝐴 +P 𝑥) = (𝐴 +P 𝑦) → 𝑥 = 𝑦)) | |
7 | 5, 6 | syl5 32 | . . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ P ∧ 𝑦 ∈ P) → (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
8 | 7 | 3expa 1139 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑥 ∈ P) ∧ 𝑦 ∈ P) → (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
9 | 8 | ralrimiva 2439 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ P) → ∀𝑦 ∈ P (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
10 | 9 | ralrimiva 2439 | . . . 4 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ P ∀𝑦 ∈ P (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
11 | oveq2 5572 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 +P 𝑥) = (𝐴 +P 𝑦)) | |
12 | 11 | eqeq1d 2091 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P 𝑦) = 𝐵)) |
13 | 12 | rmo4 2794 | . . . 4 ⊢ (∃*𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵 ↔ ∀𝑥 ∈ P ∀𝑦 ∈ P (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
14 | 10, 13 | sylibr 132 | . . 3 ⊢ (𝐴 ∈ P → ∃*𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
15 | 4, 14 | syl 14 | . 2 ⊢ (𝐴<P 𝐵 → ∃*𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
16 | reu5 2571 | . 2 ⊢ (∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵 ↔ (∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵 ∧ ∃*𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)) | |
17 | 1, 15, 16 | sylanbrc 408 | 1 ⊢ (𝐴<P 𝐵 → ∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 ∀wral 2353 ∃wrex 2354 ∃!wreu 2355 ∃*wrmo 2356 class class class wbr 3805 (class class class)co 5564 Pcnp 6613 +P cpp 6615 <P cltp 6617 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-eprel 4072 df-id 4076 df-po 4079 df-iso 4080 df-iord 4149 df-on 4151 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-1st 5819 df-2nd 5820 df-recs 5975 df-irdg 6040 df-1o 6086 df-2o 6087 df-oadd 6090 df-omul 6091 df-er 6194 df-ec 6196 df-qs 6200 df-ni 6626 df-pli 6627 df-mi 6628 df-lti 6629 df-plpq 6666 df-mpq 6667 df-enq 6669 df-nqqs 6670 df-plqqs 6671 df-mqqs 6672 df-1nqqs 6673 df-rq 6674 df-ltnqqs 6675 df-enq0 6746 df-nq0 6747 df-0nq0 6748 df-plq0 6749 df-mq0 6750 df-inp 6788 df-iplp 6790 df-iltp 6792 |
This theorem is referenced by: srpospr 7091 |
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