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Mirrors > Home > ILE Home > Th. List > ltordlem | GIF version |
Description: Lemma for eqord1 8269. (Contributed by Mario Carneiro, 14-Jun-2014.) |
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)) |
Ref | Expression |
---|---|
ltordlem | ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltord.6 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵)) | |
2 | 1 | ralrimivva 2517 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → 𝐴 < 𝐵)) |
3 | breq1 3940 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 < 𝑦 ↔ 𝐶 < 𝑦)) | |
4 | ltord.2 | . . . . 5 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) | |
5 | 4 | breq1d 3947 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 < 𝐵 ↔ 𝑀 < 𝐵)) |
6 | 3, 5 | imbi12d 233 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 < 𝑦 → 𝐴 < 𝐵) ↔ (𝐶 < 𝑦 → 𝑀 < 𝐵))) |
7 | breq2 3941 | . . . 4 ⊢ (𝑦 = 𝐷 → (𝐶 < 𝑦 ↔ 𝐶 < 𝐷)) | |
8 | eqeq1 2147 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐷 ↔ 𝑦 = 𝐷)) | |
9 | ltord.1 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
10 | 9 | eqeq1d 2149 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑁 ↔ 𝐵 = 𝑁)) |
11 | 8, 10 | imbi12d 233 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐷 → 𝐴 = 𝑁) ↔ (𝑦 = 𝐷 → 𝐵 = 𝑁))) |
12 | ltord.3 | . . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) | |
13 | 11, 12 | chvarv 1910 | . . . . 5 ⊢ (𝑦 = 𝐷 → 𝐵 = 𝑁) |
14 | 13 | breq2d 3949 | . . . 4 ⊢ (𝑦 = 𝐷 → (𝑀 < 𝐵 ↔ 𝑀 < 𝑁)) |
15 | 7, 14 | imbi12d 233 | . . 3 ⊢ (𝑦 = 𝐷 → ((𝐶 < 𝑦 → 𝑀 < 𝐵) ↔ (𝐶 < 𝐷 → 𝑀 < 𝑁))) |
16 | 6, 15 | rspc2v 2806 | . 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → 𝐴 < 𝐵) → (𝐶 < 𝐷 → 𝑀 < 𝑁))) |
17 | 2, 16 | mpan9 279 | 1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∀wral 2417 ⊆ wss 3076 class class class wbr 3937 ℝcr 7643 < clt 7824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 |
This theorem is referenced by: eqord1 8269 |
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