qextle - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > qextle		Structured version Visualization version GIF version

Theorem qextle 13231

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

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

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem qextle**
Step	Hyp	Ref	Expression
1		breq2 5114	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))
2	1	ralrimivw 3161	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℚ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))
3		xrlttri2 13168	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
4		qextltlem 13229	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))))
5		simpr 489	. . . . . . . 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 13229	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴))))
9		simpr 489	. . . . . . . . . 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 463	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)))
15	7, 14	jaod 872	. . . . 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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 class class class wbr 5110 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℚcq 12973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 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 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-sup 9403 df-inf 9404 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-n0 12506 df-z 12593 df-uz 12864 df-q 12974

This theorem is referenced by: (None)

W3C validator