![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > lteupri | GIF version |
Description: The difference from ltexpri 7600 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 7600 | . 2 ⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | |
2 | ltrelpr 7492 | . . . . 5 ⊢ <P ⊆ (P × P) | |
3 | 2 | brel 4675 | . . . 4 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
4 | 3 | simpld 112 | . . 3 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P) |
5 | eqtr3 2197 | . . . . . . . 8 ⊢ (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → (𝐴 +P 𝑥) = (𝐴 +P 𝑦)) | |
6 | addcanprg 7603 | . . . . . . . 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 1203 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑥 ∈ P) ∧ 𝑦 ∈ P) → (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
9 | 8 | ralrimiva 2550 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ P) → ∀𝑦 ∈ P (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
10 | 9 | ralrimiva 2550 | . . . 4 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ P ∀𝑦 ∈ P (((𝐴 +P 𝑥) = 𝐵 ∧ (𝐴 +P 𝑦) = 𝐵) → 𝑥 = 𝑦)) |
11 | oveq2 5877 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 +P 𝑥) = (𝐴 +P 𝑦)) | |
12 | 11 | eqeq1d 2186 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P 𝑦) = 𝐵)) |
13 | 12 | rmo4 2930 | . . . 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 134 | . . 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 2689 | . 2 ⊢ (∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵 ↔ (∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵 ∧ ∃*𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)) | |
17 | 1, 15, 16 | sylanbrc 417 | 1 ⊢ (𝐴<P 𝐵 → ∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ∃!wreu 2457 ∃*wrmo 2458 class class class wbr 4000 (class class class)co 5869 Pcnp 7278 +P cpp 7280 <P cltp 7282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4286 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-1o 6411 df-2o 6412 df-oadd 6415 df-omul 6416 df-er 6529 df-ec 6531 df-qs 6535 df-ni 7291 df-pli 7292 df-mi 7293 df-lti 7294 df-plpq 7331 df-mpq 7332 df-enq 7334 df-nqqs 7335 df-plqqs 7336 df-mqqs 7337 df-1nqqs 7338 df-rq 7339 df-ltnqqs 7340 df-enq0 7411 df-nq0 7412 df-0nq0 7413 df-plq0 7414 df-mq0 7415 df-inp 7453 df-iplp 7455 df-iltp 7457 |
This theorem is referenced by: srpospr 7770 |
Copyright terms: Public domain | W3C validator |