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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.) |
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 6037 | . . 3 ⊢ (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On) |
4 | 1, 3 | mpbir 221 | . 2 ⊢ 𝐴:dom 𝐴⟶On |
5 | issmo.2 | . . 3 ⊢ Ord 𝐵 | |
6 | ordeq 5730 | . . . 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 221 | . 2 ⊢ Ord dom 𝐴 |
9 | 2 | eleq2i 2693 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ 𝐵) |
10 | 2 | eleq2i 2693 | . . . 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 496 | . . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) |
13 | 12 | rgen2a 2977 | . 2 ⊢ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) |
14 | df-smo 7443 | . 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 1244 | 1 ⊢ Smo 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 dom cdm 5114 Ord word 5722 Oncon0 5723 ⟶wf 5884 ‘cfv 5888 Smo wsmo 7442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-in 3581 df-ss 3588 df-uni 4437 df-tr 4753 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-fn 5891 df-f 5892 df-smo 7443 |
This theorem is referenced by: iordsmo 7454 smobeth 9408 |
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