Theorem eqord1 8556

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 529	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐶 ∈ 𝑆)
2		elisset 2786	. . . . . 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 2215	. . . . . . . 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 2240	. . . 4 ⊢ ((((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) ∧ 𝑥 = 𝐶) → 𝑀 = 𝑁)
13	4, 12	exlimddv 1922	. . 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 8555	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁))
20	19	con3d 632	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (¬ 𝑀 < 𝑁 → ¬ 𝐶 < 𝐷))
21	15, 10, 5, 16, 17, 18	ltordlem 8555	. . . . . 6 ⊢ ((𝜑 ∧ (𝐷 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐷 < 𝐶 → 𝑁 < 𝑀))
22	21	con3d 632	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (¬ 𝑁 < 𝑀 → ¬ 𝐷 < 𝐶))
23	22	ancom2s 566	. . . 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 2579	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ)
26	5	eleq1d 2274	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ))
27	26	rspccva 2876	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ)
28	25, 27	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ)
29	28	adantrr 479	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑀 ∈ ℝ)
30	10	eleq1d 2274	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ))
31	30	rspccva 2876	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ)
32	25, 31	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ)
33	32	adantrl 478	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑁 ∈ ℝ)
34	29, 33	lttri3d 8187	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝑀 = 𝑁 ↔ (¬ 𝑀 < 𝑁 ∧ ¬ 𝑁 < 𝑀)))
35	16, 1	sselid 3191	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐶 ∈ ℝ)
36		simprr 531	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐷 ∈ 𝑆)
37	16, 36	sselid 3191	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐷 ∈ ℝ)
38	35, 37	lttri3d 8187	. . 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 1373  ∃wex 1515   ∈ wcel 2176  ∀wral 2484   ⊆ wss 3166   class class class wbr 4044  ℝcr 7924   < clt 8107

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltirr 8037  ax-pre-apti 8040

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-pnf 8109  df-mnf 8110  df-ltxr 8112

This theorem is referenced by:  eqord2  8557  reef11  12010  nninfdclemf1  12823