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Theorem enqeq 9635

Description: Corollary of nqereu 9630: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
enqeq	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐴 = 𝐵)

Proof of Theorem enqeq Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpa 1051	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
2		elpqn 9626	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
3	2	3ad2ant2 1076	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ (N × N))
4		nqereu 9630	. . . 4 ⊢ (𝐵 ∈ (N × N) → ∃!𝑥 ∈ Q 𝑥 ~_Q 𝐵)
5		reurmo 3138	. . . 4 ⊢ (∃!𝑥 ∈ Q 𝑥 ~_Q 𝐵 → ∃*𝑥 ∈ Q 𝑥 ~_Q 𝐵)
6	3, 4, 5	3syl 18	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ∃*𝑥 ∈ Q 𝑥 ~_Q 𝐵)
7		df-rmo 2904	. . 3 ⊢ (∃𝑥 ∈ Q 𝑥 ~_Q 𝐵 ↔ ∃𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
8	6, 7	sylib 207	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
9		3simpb 1052	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵))
10		simp2 1055	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ Q)
11		enqer 9622	. . . . 5 ⊢ ~_Q Er (N × N)
12	11	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ~_Q Er (N × N))
13	12, 3	erref 7649	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ~_Q 𝐵)
14	10, 13	jca 553	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵))
15		eleq1 2676	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Q ↔ 𝐴 ∈ Q))
16		breq1 4586	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ~_Q 𝐵 ↔ 𝐴 ~_Q 𝐵))
17	15, 16	anbi12d 743	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ↔ (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵)))
18		eleq1 2676	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ Q ↔ 𝐵 ∈ Q))
19		breq1 4586	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ~_Q 𝐵 ↔ 𝐵 ~_Q 𝐵))
20	18, 19	anbi12d 743	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ↔ (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵)))
21	17, 20	moi 3356	. 2 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ∧ ((𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵) ∧ (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵))) → 𝐴 = 𝐵)
22	1, 8, 9, 14, 21	syl112anc 1322	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐴 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 ∃!wreu 2898 ∃*wrmo 2899 class class class wbr 4583 × cxp 5036 Er wer 7626 Ncnpi 9545 ~_Q ceq 9552 Qcnq 9553

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-omul 7452 df-er 7629 df-ni 9573 df-mi 9575 df-lti 9576 df-enq 9612 df-nq 9613

This theorem is referenced by: nqereq 9636 ltsonq 9670

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