nqprlu - Intuitionistic Logic Explorer

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Theorem nqprlu 7577

Description: The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)

**Assertion**
Ref	Expression
nqprlu	⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ∈ P)

Distinct variable groups: 𝐴,𝑙 𝑢,𝐴

Proof of Theorem nqprlu Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4022	. . . . 5 ⊢ (𝑙 = 𝑎 → (𝐴 <_Q 𝑙 ↔ 𝐴 <_Q 𝑎))
2	1	cbvabv 2314	. . . 4 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑎 ∣ 𝐴 <_Q 𝑎}
3		breq2 4022	. . . . 5 ⊢ (𝑢 = 𝑎 → (𝐴 <_Q 𝑢 ↔ 𝐴 <_Q 𝑎))
4	3	cbvabv 2314	. . . 4 ⊢ {𝑢 ∣ 𝐴 <_Q 𝑢} = {𝑎 ∣ 𝐴 <_Q 𝑎}
5	2, 4	eqtr4i 2213	. . 3 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑢 ∣ 𝐴 <_Q 𝑢}
6	5	opeq2i 3797	. 2 ⊢ ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ = ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩
7		nqprxx 7576	. 2 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ ∈ P)
8	6, 7	eqeltrrid 2277	1 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ∈ P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2160 {cab 2175 ⟨cop 3610 class class class wbr 4018 Qcnq 7310 <_Q cltq 7315 Pcnp 7321

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-1o 6442 df-oadd 6446 df-omul 6447 df-er 6560 df-ec 6562 df-qs 6566 df-ni 7334 df-pli 7335 df-mi 7336 df-lti 7337 df-plpq 7374 df-mpq 7375 df-enq 7377 df-nqqs 7378 df-plqqs 7379 df-mqqs 7380 df-1nqqs 7381 df-rq 7382 df-ltnqqs 7383 df-inp 7496