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Theorem enqeq 10843

Description: Corollary of nqereu 10838: 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 1148	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
2		elpqn 10834	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
3	2	3ad2ant2 1134	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ (N × N))
4		nqereu 10838	. . . 4 ⊢ (𝐵 ∈ (N × N) → ∃!𝑥 ∈ Q 𝑥 ~_Q 𝐵)
5		reurmo 3351	. . . 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 3348	. . 3 ⊢ (∃𝑥 ∈ Q 𝑥 ~_Q 𝐵 ↔ ∃𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
8	6, 7	sylib 218	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
9		3simpb 1149	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵))
10		simp2 1137	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ Q)
11		enqer 10830	. . . . 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 8653	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ~_Q 𝐵)
14	10, 13	jca 511	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵))
15		eleq1 2822	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Q ↔ 𝐴 ∈ Q))
16		breq1 5099	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ~_Q 𝐵 ↔ 𝐴 ~_Q 𝐵))
17	15, 16	anbi12d 632	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ↔ (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵)))
18		eleq1 2822	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ Q ↔ 𝐵 ∈ Q))
19		breq1 5099	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ~_Q 𝐵 ↔ 𝐵 ~_Q 𝐵))
20	18, 19	anbi12d 632	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ↔ (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵)))
21	17, 20	moi 3674	. 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 1376	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐴 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∃*wmo 2535 ∃!wreu 3346 ∃*wrmo 3347 class class class wbr 5096 × cxp 5620 Er wer 8630 Ncnpi 10753 ~_Q ceq 10760 Qcnq 10761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-oadd 8399 df-omul 8400 df-er 8633 df-ni 10781 df-mi 10783 df-lti 10784 df-enq 10820 df-nq 10821

This theorem is referenced by: nqereq 10844 ltsonq 10878

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