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| Mirrors > Home > MPE Home > Th. List > issmo | Structured version Visualization version GIF version | ||
| Description: Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) Avoid ax-13 2377. (Revised by GG, 19-May-2023.) |
| Ref | Expression |
|---|---|
| issmo.1 | ⊢ 𝐴:𝐵⟶On |
| issmo.2 | ⊢ Ord 𝐵 |
| issmo.3 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) |
| issmo.4 | ⊢ dom 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| issmo | ⊢ Smo 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issmo.1 | . . 3 ⊢ 𝐴:𝐵⟶On | |
| 2 | issmo.4 | . . . 4 ⊢ dom 𝐴 = 𝐵 | |
| 3 | 2 | feq2i 6728 | . . 3 ⊢ (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On) |
| 4 | 1, 3 | mpbir 231 | . 2 ⊢ 𝐴:dom 𝐴⟶On |
| 5 | issmo.2 | . . 3 ⊢ Ord 𝐵 | |
| 6 | ordeq 6391 | . . . 4 ⊢ (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵)) | |
| 7 | 2, 6 | ax-mp 5 | . . 3 ⊢ (Ord dom 𝐴 ↔ Ord 𝐵) |
| 8 | 5, 7 | mpbir 231 | . 2 ⊢ Ord dom 𝐴 |
| 9 | 2 | eleq2i 2833 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ 𝐵) |
| 10 | 2 | eleq2i 2833 | . . . 4 ⊢ (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ 𝐵) |
| 11 | issmo.3 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) | |
| 12 | 9, 10, 11 | syl2anb 598 | . . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) |
| 13 | 12 | rgen2 3199 | . 2 ⊢ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) |
| 14 | df-smo 8386 | . 2 ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))) | |
| 15 | 4, 8, 13, 14 | mpbir3an 1342 | 1 ⊢ Smo 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 dom cdm 5685 Ord word 6383 Oncon0 6384 ⟶wf 6557 ‘cfv 6561 Smo wsmo 8385 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-fn 6564 df-f 6565 df-smo 8386 |
| This theorem is referenced by: iordsmo 8397 smobeth 10626 |
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