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Theorem issmo 5844
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 ((x B y B) → (x y → (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 4983 . . 3 (A:dom A⟶On ↔ A:B⟶On)
41, 3mpbir 134 . 2 A:dom A⟶On
5 issmo.2 . . 3 Ord B
6 ordeq 4075 . . . 4 (dom A = B → (Ord dom A ↔ Ord B))
72, 6ax-mp 7 . . 3 (Ord dom A ↔ Ord B)
85, 7mpbir 134 . 2 Ord dom A
92eleq2i 2101 . . . 4 (x dom Ax B)
102eleq2i 2101 . . . 4 (y dom Ay B)
11 issmo.3 . . . 4 ((x B y B) → (x y → (Ax) (Ay)))
129, 10, 11syl2anb 275 . . 3 ((x dom A y dom A) → (x y → (Ax) (Ay)))
1312rgen2a 2369 . 2 x dom Ay dom A(x y → (Ax) (Ay))
14 df-smo 5842 . 2 (Smo A ↔ (A:dom A⟶On Ord dom A x dom Ay dom A(x y → (Ax) (Ay))))
154, 8, 13, 14mpbir3an 1085 1 Smo A
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  Ord word 4065  Oncon0 4066  dom cdm 4288  wf 4841  cfv 4845  Smo wsmo 5841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-fn 4848  df-f 4849  df-smo 5842
This theorem is referenced by:  iordsmo  5853
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