Theorem eqord1 8801

Description: A strictly increasing real function on a subset of ℝ is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)

Hypotheses

Ref	Expression
ltord.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
ltord.2	⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)
ltord.3	⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)
ltord.4	⊢ 𝑆 ⊆ ℝ
ltord.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
ltord.6	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵))

Assertion

Ref	Expression
eqord1	⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem eqord1

Step	Hyp	Ref	Expression
1		simprl 535	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐶 ∈ 𝑆)
2		elisset 2836	. . . . . 6 ⊢ (𝐶 ∈ 𝑆 → ∃𝑥 𝑥 = 𝐶)
3	1, 2	syl 14	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ∃𝑥 𝑥 = 𝐶)
4	3	adantr 276	. . . 4 ⊢ (((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) → ∃𝑥 𝑥 = 𝐶)
5		ltord.2	. . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)
6	5	adantl 277	. . . . 5 ⊢ ((((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) ∧ 𝑥 = 𝐶) → 𝐴 = 𝑀)
7		eqeq2 2248	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → (𝑥 = 𝐶 ↔ 𝑥 = 𝐷))
8	7	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) → (𝑥 = 𝐶 ↔ 𝑥 = 𝐷))
9	8	biimpa 296	. . . . . 6 ⊢ ((((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) ∧ 𝑥 = 𝐶) → 𝑥 = 𝐷)
10		ltord.3	. . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)
11	9, 10	syl 14	. . . . 5 ⊢ ((((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) ∧ 𝑥 = 𝐶) → 𝐴 = 𝑁)
12	6, 11	eqtr3d 2273	. . . 4 ⊢ ((((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) ∧ 𝑥 = 𝐶) → 𝑀 = 𝑁)
13	4, 12	exlimddv 1954	. . 3 ⊢ (((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) → 𝑀 = 𝑁)
14	13	ex 115	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 → 𝑀 = 𝑁))
15		ltord.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
16		ltord.4	. . . . . 6 ⊢ 𝑆 ⊆ ℝ
17		ltord.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
18		ltord.6	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵))
19	15, 5, 10, 16, 17, 18	ltordlem 8800	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁))
20	19	con3d 640	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (¬ 𝑀 < 𝑁 → ¬ 𝐶 < 𝐷))
21	15, 10, 5, 16, 17, 18	ltordlem 8800	. . . . . 6 ⊢ ((𝜑 ∧ (𝐷 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐷 < 𝐶 → 𝑁 < 𝑀))
22	21	con3d 640	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (¬ 𝑁 < 𝑀 → ¬ 𝐷 < 𝐶))
23	22	ancom2s 572	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (¬ 𝑁 < 𝑀 → ¬ 𝐷 < 𝐶))
24	20, 23	anim12d 335	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ((¬ 𝑀 < 𝑁 ∧ ¬ 𝑁 < 𝑀) → (¬ 𝐶 < 𝐷 ∧ ¬ 𝐷 < 𝐶)))
25	17	ralrimiva 2623	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ)
26	5	eleq1d 2307	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ))
27	26	rspccva 2928	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ)
28	25, 27	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ)
29	28	adantrr 483	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑀 ∈ ℝ)
30	10	eleq1d 2307	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ))
31	30	rspccva 2928	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ)
32	25, 31	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ)
33	32	adantrl 482	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑁 ∈ ℝ)
34	29, 33	lttri3d 8430	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝑀 = 𝑁 ↔ (¬ 𝑀 < 𝑁 ∧ ¬ 𝑁 < 𝑀)))
35	16, 1	sselid 3246	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐶 ∈ ℝ)
36		simprr 537	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐷 ∈ 𝑆)
37	16, 36	sselid 3246	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐷 ∈ ℝ)
38	35, 37	lttri3d 8430	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ (¬ 𝐶 < 𝐷 ∧ ¬ 𝐷 < 𝐶)))
39	24, 34, 38	3imtr4d 203	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝑀 = 𝑁 → 𝐶 = 𝐷))
40	14, 39	impbid 129	1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220   class class class wbr 4125  ℝcr 8168   < clt 8350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281  ax-pre-apti 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-pnf 8352  df-mnf 8353  df-ltxr 8355

This theorem is referenced by:  eqord2  8802  reef11  12444  nninfdclemf1  13321