nqprlu - Intuitionistic Logic Explorer

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Theorem nqprlu 7614

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 4037	. . . . 5 ⊢ (𝑙 = 𝑎 → (𝐴 <_Q 𝑙 ↔ 𝐴 <_Q 𝑎))
2	1	cbvabv 2321	. . . 4 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑎 ∣ 𝐴 <_Q 𝑎}
3		breq2 4037	. . . . 5 ⊢ (𝑢 = 𝑎 → (𝐴 <_Q 𝑢 ↔ 𝐴 <_Q 𝑎))
4	3	cbvabv 2321	. . . 4 ⊢ {𝑢 ∣ 𝐴 <_Q 𝑢} = {𝑎 ∣ 𝐴 <_Q 𝑎}
5	2, 4	eqtr4i 2220	. . 3 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑢 ∣ 𝐴 <_Q 𝑢}
6	5	opeq2i 3812	. 2 ⊢ ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ = ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩
7		nqprxx 7613	. 2 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ ∈ P)
8	6, 7	eqeltrrid 2284	1 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ∈ P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2167 {cab 2182 ⟨cop 3625 class class class wbr 4033 Qcnq 7347 <_Q cltq 7352 Pcnp 7358

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-eprel 4324 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-1o 6474 df-oadd 6478 df-omul 6479 df-er 6592 df-ec 6594 df-qs 6598 df-ni 7371 df-pli 7372 df-mi 7373 df-lti 7374 df-plpq 7411 df-mpq 7412 df-enq 7414 df-nqqs 7415 df-plqqs 7416 df-mqqs 7417 df-1nqqs 7418 df-rq 7419 df-ltnqqs 7420 df-inp 7533