nnnq - Intuitionistic Logic Explorer

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Theorem nnnq 6612

Description: The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)

**Assertion**
Ref	Expression
nnnq	⊢ (𝐴 ∈ N → [⟨𝐴, 1_𝑜⟩] ~_Q ∈ Q)

**Proof of Theorem nnnq**
Step	Hyp	Ref	Expression
1		1pi 6505	. . . 4 ⊢ 1_𝑜 ∈ N
2		opelxpi 4394	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1_𝑜 ∈ N) → ⟨𝐴, 1_𝑜⟩ ∈ (N × N))
3	1, 2	mpan2 415	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1_𝑜⟩ ∈ (N × N))
4		enqex 6550	. . . 4 ⊢ ~_Q ∈ V
5	4	ecelqsi 6183	. . 3 ⊢ (⟨𝐴, 1_𝑜⟩ ∈ (N × N) → [⟨𝐴, 1_𝑜⟩] ~_Q ∈ ((N × N) / ~_Q ))
6	3, 5	syl 14	. 2 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_𝑜⟩] ~_Q ∈ ((N × N) / ~_Q ))
7		df-nqqs 6538	. 2 ⊢ Q = ((N × N) / ~_Q )
8	6, 7	syl6eleqr 2172	1 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_𝑜⟩] ~_Q ∈ Q)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 1433 ⟨cop 3401 × cxp 4361 1_𝑜c1o 6017 [cec 6127 / cqs 6128 Ncnpi 6462 ~_Q ceq 6469 Qcnq 6470

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-suc 4126 df-iom 4332 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-1o 6024 df-ec 6131 df-qs 6135 df-ni 6494 df-enq 6537 df-nqqs 6538