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Theorem qextlt 13343

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 5109	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
2	1	ralrimivw 3161	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
3		xrlttri2 13281	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
4		qextltlem 13342	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))))
5		simpl 488	. . . . . . . 8 ⊢ ((¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)) → ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
6	5	reximi 3103	. . . . . . 7 ⊢ (∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)) → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
7	4, 6	syl6 36	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
8		qextltlem 13342	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴))))
9		simpl 488	. . . . . . . . . 10 ⊢ ((¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴)) → ¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴))
10		bicom 225	. . . . . . . . . 10 ⊢ ((𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ↔ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
11	9, 10	sylnib 331	. . . . . . . . 9 ⊢ ((¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴)) → ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
12	11	reximi 3103	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴)) → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
13	8, 12	syl6 36	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
14	13	ancoms 464	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
15	7, 14	jaod 873	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
16	3, 15	sylbid 243	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 → ∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
17		rexnal 3117	. . . 4 ⊢ (∃𝑥 ∈ ℚ ¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ↔ ¬ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))
18	16, 17	imbitrdi 254	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 → ¬ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))
19	18	necon4ad 2977	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) → 𝐴 = 𝐵))
20	2, 19	impbid2 229	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2146 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 class class class wbr 5105 ℝ^*cxr 11352 < clt 11353 ≤ cle 11354 ℚcq 13085

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 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 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9444 df-inf 9445 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-n0 12617 df-z 12704 df-uz 12976 df-q 13086

This theorem is used by: (None)

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