nqprlu - Intuitionistic Logic Explorer

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Theorem nqprlu 7767

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 4092	. . . . 5 ⊢ (𝑙 = 𝑎 → (𝐴 <_Q 𝑙 ↔ 𝐴 <_Q 𝑎))
2	1	cbvabv 2356	. . . 4 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑎 ∣ 𝐴 <_Q 𝑎}
3		breq2 4092	. . . . 5 ⊢ (𝑢 = 𝑎 → (𝐴 <_Q 𝑢 ↔ 𝐴 <_Q 𝑎))
4	3	cbvabv 2356	. . . 4 ⊢ {𝑢 ∣ 𝐴 <_Q 𝑢} = {𝑎 ∣ 𝐴 <_Q 𝑎}
5	2, 4	eqtr4i 2255	. . 3 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑢 ∣ 𝐴 <_Q 𝑢}
6	5	opeq2i 3866	. 2 ⊢ ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ = ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩
7		nqprxx 7766	. 2 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ ∈ P)
8	6, 7	eqeltrrid 2319	1 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ∈ P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2202 {cab 2217 ⟨cop 3672 class class class wbr 4088 Qcnq 7500 <_Q cltq 7505 Pcnp 7511

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-eprel 4386 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-1o 6582 df-oadd 6586 df-omul 6587 df-er 6702 df-ec 6704 df-qs 6708 df-ni 7524 df-pli 7525 df-mi 7526 df-lti 7527 df-plpq 7564 df-mpq 7565 df-enq 7567 df-nqqs 7568 df-plqqs 7569 df-mqqs 7570 df-1nqqs 7571 df-rq 7572 df-ltnqqs 7573 df-inp 7686