nqprlu - Intuitionistic Logic Explorer

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Theorem nqprlu 7697

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 4064	. . . . 5 ⊢ (𝑙 = 𝑎 → (𝐴 <_Q 𝑙 ↔ 𝐴 <_Q 𝑎))
2	1	cbvabv 2332	. . . 4 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑎 ∣ 𝐴 <_Q 𝑎}
3		breq2 4064	. . . . 5 ⊢ (𝑢 = 𝑎 → (𝐴 <_Q 𝑢 ↔ 𝐴 <_Q 𝑎))
4	3	cbvabv 2332	. . . 4 ⊢ {𝑢 ∣ 𝐴 <_Q 𝑢} = {𝑎 ∣ 𝐴 <_Q 𝑎}
5	2, 4	eqtr4i 2231	. . 3 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑢 ∣ 𝐴 <_Q 𝑢}
6	5	opeq2i 3838	. 2 ⊢ ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ = ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩
7		nqprxx 7696	. 2 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ ∈ P)
8	6, 7	eqeltrrid 2295	1 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ∈ P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2178 {cab 2193 ⟨cop 3647 class class class wbr 4060 Qcnq 7430 <_Q cltq 7435 Pcnp 7441

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4176 ax-sep 4179 ax-nul 4187 ax-pow 4235 ax-pr 4270 ax-un 4499 ax-setind 4604 ax-iinf 4655

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2779 df-sbc 3007 df-csb 3103 df-dif 3177 df-un 3179 df-in 3181 df-ss 3188 df-nul 3470 df-pw 3629 df-sn 3650 df-pr 3651 df-op 3653 df-uni 3866 df-int 3901 df-iun 3944 df-br 4061 df-opab 4123 df-mpt 4124 df-tr 4160 df-eprel 4355 df-id 4359 df-po 4362 df-iso 4363 df-iord 4432 df-on 4434 df-suc 4437 df-iom 4658 df-xp 4700 df-rel 4701 df-cnv 4702 df-co 4703 df-dm 4704 df-rn 4705 df-res 4706 df-ima 4707 df-iota 5252 df-fun 5293 df-fn 5294 df-f 5295 df-f1 5296 df-fo 5297 df-f1o 5298 df-fv 5299 df-ov 5972 df-oprab 5973 df-mpo 5974 df-1st 6251 df-2nd 6252 df-recs 6416 df-irdg 6481 df-1o 6527 df-oadd 6531 df-omul 6532 df-er 6645 df-ec 6647 df-qs 6651 df-ni 7454 df-pli 7455 df-mi 7456 df-lti 7457 df-plpq 7494 df-mpq 7495 df-enq 7497 df-nqqs 7498 df-plqqs 7499 df-mqqs 7500 df-1nqqs 7501 df-rq 7502 df-ltnqqs 7503 df-inp 7616