nnnq - Intuitionistic Logic Explorer

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Theorem nnnq 7642

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 7535	. . . 4 ⊢ 1_o ∈ N
2		opelxpi 4757	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1_o ∈ N) → ⟨𝐴, 1_o⟩ ∈ (N × N))
3	1, 2	mpan2 425	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1_o⟩ ∈ (N × N))
4		enqex 7580	. . . 4 ⊢ ~_Q ∈ V
5	4	ecelqsi 6758	. . 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 7568	. 2 ⊢ Q = ((N × N) / ~_Q )
8	6, 7	eleqtrrdi 2325	1 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ Q)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2202 ⟨cop 3672 × cxp 4723 1_oc1o 6575 [cec 6700 / cqs 6701 Ncnpi 7492 ~_Q ceq 7499 Qcnq 7500

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-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 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-v 2804 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-br 4089 df-opab 4151 df-suc 4468 df-iom 4689 df-xp 4731 df-cnv 4733 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-1o 6582 df-ec 6704 df-qs 6708 df-ni 7524 df-enq 7567 df-nqqs 7568