nqprlu - Intuitionistic Logic Explorer

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Theorem nqprlu 7546

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 4008	. . . . 5 ⊢ (𝑙 = 𝑎 → (𝐴 <_Q 𝑙 ↔ 𝐴 <_Q 𝑎))
2	1	cbvabv 2302	. . . 4 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑎 ∣ 𝐴 <_Q 𝑎}
3		breq2 4008	. . . . 5 ⊢ (𝑢 = 𝑎 → (𝐴 <_Q 𝑢 ↔ 𝐴 <_Q 𝑎))
4	3	cbvabv 2302	. . . 4 ⊢ {𝑢 ∣ 𝐴 <_Q 𝑢} = {𝑎 ∣ 𝐴 <_Q 𝑎}
5	2, 4	eqtr4i 2201	. . 3 ⊢ {𝑙 ∣ 𝐴 <_Q 𝑙} = {𝑢 ∣ 𝐴 <_Q 𝑢}
6	5	opeq2i 3783	. 2 ⊢ ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ = ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩
7		nqprxx 7545	. 2 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑙 ∣ 𝐴 <_Q 𝑙}⟩ ∈ P)
8	6, 7	eqeltrrid 2265	1 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ∈ P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2148 {cab 2163 ⟨cop 3596 class class class wbr 4004 Qcnq 7279 <_Q cltq 7284 Pcnp 7290

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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588

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-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-eprel 4290 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-1o 6417 df-oadd 6421 df-omul 6422 df-er 6535 df-ec 6537 df-qs 6541 df-ni 7303 df-pli 7304 df-mi 7305 df-lti 7306 df-plpq 7343 df-mpq 7344 df-enq 7346 df-nqqs 7347 df-plqqs 7348 df-mqqs 7349 df-1nqqs 7350 df-rq 7351 df-ltnqqs 7352 df-inp 7465