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Theorem qextle 13131

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 5104	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))
2	1	ralrimivw 3134	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℚ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))
3		xrlttri2 13068	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
4		qextltlem 13129	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))))
5		simpr 484	. . . . . . . 8 ⊢ ((¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)) → ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))
6	5	reximi 3076	. . . . . . 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 13129	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐵 ↔ 𝑥 < 𝐴) ∧ ¬ (𝑥 ≤ 𝐵 ↔ 𝑥 ≤ 𝐴))))
9		simpr 484	. . . . . . . . . 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 3076	. . . . . . . 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 860	. . . . 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 3090	. . . 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 2952	. 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 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 class class class wbr 5100 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 ℚcq 12873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874

This theorem is referenced by: (None)

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