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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 2375. (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 6729 | . . 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 6393 | . . . 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 2831 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ 𝐵) |
10 | 2 | eleq2i 2831 | . . . 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 3197 | . 2 ⊢ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) |
14 | df-smo 8385 | . 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 1340 | 1 ⊢ Smo 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 dom cdm 5689 Ord word 6385 Oncon0 6386 ⟶wf 6559 ‘cfv 6563 Smo wsmo 8384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-ss 3980 df-uni 4913 df-tr 5266 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-fn 6566 df-f 6567 df-smo 8385 |
This theorem is referenced by: iordsmo 8396 smobeth 10624 |
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