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Theorem qextlt 13219

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 5123	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
2	1	ralrimivw 3136	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
3		xrlttri2 13158	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
4		qextltlem 13218	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))))
5		simpl 482	. . . . . . . 8 ⊢ ((¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)) → ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
6	5	reximi 3074	. . . . . . 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 13218	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴))))
9		simpl 482	. . . . . . . . . 10 ⊢ ((¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴)) → ¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴))
10		bicom 222	. . . . . . . . . 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 3074	. . . . . . . 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 458	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
15	7, 14	jaod 859	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
16	3, 15	sylbid 240	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
17		rexnal 3089	. . . 4 ⊢ (∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ↔ ¬ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
18	16, 17	imbitrdi 251	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 → ¬ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
19	18	necon4ad 2951	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) → 𝐴 = 𝐵))
20	2, 19	impbid2 226	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 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 class class class wbr 5119 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 ℚcq 12964

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853 df-q 12965

This theorem is referenced by: (None)

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