1nq - Intuitionistic Logic Explorer

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Theorem 1nq 6464

Description: The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)

**Assertion**
Ref	Expression
1nq	⊢ 1_Q ∈ Q

**Proof of Theorem 1nq**
Step	Hyp	Ref	Expression
1		1pi 6413	. . . 4 ⊢ 1_𝑜 ∈ N
2		opelxpi 4376	. . . 4 ⊢ ((1_𝑜 ∈ N ∧ 1_𝑜 ∈ N) → ⟨1_𝑜, 1_𝑜⟩ ∈ (N × N))
3	1, 1, 2	mp2an 402	. . 3 ⊢ ⟨1_𝑜, 1_𝑜⟩ ∈ (N × N)
4		enqex 6458	. . . 4 ⊢ ~_Q ∈ V
5	4	ecelqsi 6160	. . 3 ⊢ (⟨1_𝑜, 1_𝑜⟩ ∈ (N × N) → [⟨1_𝑜, 1_𝑜⟩] ~_Q ∈ ((N × N) / ~_Q ))
6	3, 5	ax-mp 7	. 2 ⊢ [⟨1_𝑜, 1_𝑜⟩] ~_Q ∈ ((N × N) / ~_Q )
7		df-1nqqs 6449	. 2 ⊢ 1_Q = [⟨1_𝑜, 1_𝑜⟩] ~_Q
8		df-nqqs 6446	. 2 ⊢ Q = ((N × N) / ~_Q )
9	6, 7, 8	3eltr4i 2119	1 ⊢ 1_Q ∈ Q

Colors of variables: wff set class

Syntax hints: ∈ wcel 1393 ⟨cop 3378 × cxp 4343 1_𝑜c1o 5994 [cec 6104 / cqs 6105 Ncnpi 6370 ~_Q ceq 6377 Qcnq 6378 1_Qc1q 6379

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-suc 4108 df-iom 4314 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-1o 6001 df-ec 6108 df-qs 6112 df-ni 6402 df-enq 6445 df-nqqs 6446 df-1nqqs 6449

This theorem is referenced by: recmulnqg 6489 rec1nq 6493 ltaddnq 6505 halfnqq 6508 addnqprllem 6625 addnqprulem 6626 1pr 6652 addnqpr1 6660 appdivnq 6661 1idprl 6688 1idpru 6689 recexprlemm 6722 recexprlem1ssl 6731 recexprlem1ssu 6732 cauappcvgprlemm 6743 caucvgprlemm 6766 caucvgprprlemmu 6793