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Mirrors > Home > ILE Home > Th. List > lteupri | GIF version |
Description: The difference from ltexpri 6803 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 6803 | . 2 ⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | |
2 | ltrelpr 6695 | . . . . 5 ⊢ <P ⊆ (P × P) | |
3 | 2 | brel 4410 | . . . 4 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
4 | 3 | simpld 110 | . . 3 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P) |
5 | eqtr3 2100 | . . . . . . . 8 ⊢ (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → (𝐴 +P 𝑥) = (𝐴 +P 𝑦)) | |
6 | addcanprg 6806 | . . . . . . . 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 1138 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑥 ∈ P) ∧ 𝑦 ∈ P) → (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
9 | 8 | ralrimiva 2434 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ P) → ∀𝑦 ∈ P (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
10 | 9 | ralrimiva 2434 | . . . 4 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ P ∀𝑦 ∈ P (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
11 | oveq2 5540 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 +P 𝑥) = (𝐴 +P 𝑦)) | |
12 | 11 | eqeq1d 2089 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P 𝑦) = 𝐵)) |
13 | 12 | rmo4 2785 | . . . 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 2566 | . 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 919 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 ∃!wreu 2350 ∃*wrmo 2351 class class class wbr 3785 (class class class)co 5532 Pcnp 6481 +P cpp 6483 <P cltp 6485 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658 df-iltp 6660 |
This theorem is referenced by: srpospr 6959 |
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