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Theorem issmo 6053
Description: Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
Hypotheses
Ref Expression
issmo.1 𝐴:𝐵⟶On
issmo.2 Ord 𝐵
issmo.3 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
issmo.4 dom 𝐴 = 𝐵
Assertion
Ref Expression
issmo Smo 𝐴
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem issmo
StepHypRef Expression
1 issmo.1 . . 3 𝐴:𝐵⟶On
2 issmo.4 . . . 4 dom 𝐴 = 𝐵
32feq2i 5155 . . 3 (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
41, 3mpbir 144 . 2 𝐴:dom 𝐴⟶On
5 issmo.2 . . 3 Ord 𝐵
6 ordeq 4199 . . . 4 (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵))
72, 6ax-mp 7 . . 3 (Ord dom 𝐴 ↔ Ord 𝐵)
85, 7mpbir 144 . 2 Ord dom 𝐴
92eleq2i 2154 . . . 4 (𝑥 ∈ dom 𝐴𝑥𝐵)
102eleq2i 2154 . . . 4 (𝑦 ∈ dom 𝐴𝑦𝐵)
11 issmo.3 . . . 4 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
129, 10, 11syl2anb 285 . . 3 ((𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
1312rgen2a 2429 . 2 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))
14 df-smo 6051 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
154, 8, 13, 14mpbir3an 1125 1 Smo 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  Ord word 4189  Oncon0 4190  dom cdm 4438  wf 5011  cfv 5015  Smo wsmo 6050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-fn 5018  df-f 5019  df-smo 6051
This theorem is referenced by:  iordsmo  6062
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