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Theorem omsmo 8270
Description: A strictly monotonic ordinal function on the set of natural numbers is one-to-one. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)
Assertion
Ref Expression
omsmo (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1𝐴)
Distinct variable group:   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem omsmo
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 765 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴)
2 omsmolem 8269 . . . . . . . . 9 (𝑧 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧))))
32adantl 482 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧))))
43imp 407 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧)))
5 omsmolem 8269 . . . . . . . . 9 (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
65adantr 481 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
76imp 407 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
84, 7orim12d 958 . . . . . 6 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
98ancoms 459 . . . . 5 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
109con3d 155 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦)) → ¬ (𝑦𝑧𝑧𝑦)))
11 ffvelrn 6841 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ 𝐴)
12 ssel 3958 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝐹𝑦) ∈ 𝐴 → (𝐹𝑦) ∈ On))
1311, 12syl5 34 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ On))
1413expdimp 453 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → (𝐹𝑦) ∈ On))
15 eloni 6194 . . . . . . . . 9 ((𝐹𝑦) ∈ On → Ord (𝐹𝑦))
1614, 15syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹𝑦)))
17 ffvelrn 6841 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ 𝐴)
18 ssel 3958 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐹𝑧) ∈ On))
1917, 18syl5 34 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ On))
2019expdimp 453 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → (𝐹𝑧) ∈ On))
21 eloni 6194 . . . . . . . . 9 ((𝐹𝑧) ∈ On → Ord (𝐹𝑧))
2220, 21syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → Ord (𝐹𝑧)))
2316, 22anim12d 608 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧))))
2423imp 407 . . . . . 6 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧)))
25 ordtri3 6220 . . . . . 6 ((Ord (𝐹𝑦) ∧ Ord (𝐹𝑧)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
2624, 25syl 17 . . . . 5 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
2726adantlr 711 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
28 nnord 7577 . . . . . 6 (𝑦 ∈ ω → Ord 𝑦)
29 nnord 7577 . . . . . 6 (𝑧 ∈ ω → Ord 𝑧)
30 ordtri3 6220 . . . . . 6 ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3128, 29, 30syl2an 595 . . . . 5 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3231adantl 482 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3310, 27, 323imtr4d 295 . . 3 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
3433ralrimivva 3188 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
35 dff13 7004 . 2 (𝐹:ω–1-1𝐴 ↔ (𝐹:ω⟶𝐴 ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
361, 34, 35sylanbrc 583 1 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  wss 3933  Ord word 6183  Oncon0 6184  suc csuc 6186  wf 6344  1-1wf1 6345  cfv 6348  ωcom 7569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fv 6356  df-om 7570
This theorem is referenced by:  unblem4  8761
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