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Theorem qextlt 12937

Description: An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)

**Assertion**
Ref	Expression
qextlt	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem qextlt**
Step	Hyp	Ref	Expression
1		breq2 5078	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
2	1	ralrimivw 3104	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
3		xrlttri2 12876	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
4		qextltlem 12936	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))))
5		simpl 483	. . . . . . . 8 ⊢ ((¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)) → ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
6	5	reximi 3178	. . . . . . 7 ⊢ (∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)) → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
7	4, 6	syl6 35	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
8		qextltlem 12936	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴))))
9		simpl 483	. . . . . . . . . 10 ⊢ ((¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴)) → ¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴))
10		bicom 221	. . . . . . . . . 10 ⊢ ((𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ↔ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
11	9, 10	sylnib 328	. . . . . . . . 9 ⊢ ((¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴)) → ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
12	11	reximi 3178	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴)) → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
13	8, 12	syl6 35	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
14	13	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
15	7, 14	jaod 856	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
16	3, 15	sylbid 239	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
17		rexnal 3169	. . . 4 ⊢ (∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ↔ ¬ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
18	16, 17	syl6ib 250	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 → ¬ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
19	18	necon4ad 2962	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) → 𝐴 = 𝐵))
20	2, 19	impbid2 225	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 class class class wbr 5074 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 ℚcq 12688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689

This theorem is referenced by: (None)

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