Theorem eqord1 8591

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 2791	. . . . . 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 2217	. . . . . . . 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 2242	. . . 4 ⊢ ((((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) ∧ 𝐶 = 𝐷) ∧ 𝑥 = 𝐶) → 𝑀 = 𝑁)
13	4, 12	exlimddv 1923	. . 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 8590	. . . . 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 8590	. . . . . 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 2581	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ)
26	5	eleq1d 2276	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ))
27	26	rspccva 2883	. . . . . 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 2276	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ))
31	30	rspccva 2883	. . . . . 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 8222	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝑀 = 𝑁 ↔ (¬ 𝑀 < 𝑁 ∧ ¬ 𝑁 < 𝑀)))
35	16, 1	sselid 3199	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐶 ∈ ℝ)
36		simprr 531	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐷 ∈ 𝑆)
37	16, 36	sselid 3199	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝐷 ∈ ℝ)
38	35, 37	lttri3d 8222	. . 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 1516   ∈ wcel 2178  ∀wral 2486   ⊆ wss 3174   class class class wbr 4059  ℝcr 7959   < clt 8142

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-pre-ltirr 8072  ax-pre-apti 8075

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-pnf 8144  df-mnf 8145  df-ltxr 8147

This theorem is referenced by:  eqord2  8592  reef11  12125  nninfdclemf1  12938