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Theorem omsmo 8679
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 767 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴)
2 omsmolem 8678 . . . . . . . . 9 (𝑧 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧))))
32adantl 480 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧))))
43imp 405 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑦𝑧 → (𝐹𝑦) ∈ (𝐹𝑧)))
5 omsmolem 8678 . . . . . . . . 9 (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
65adantr 479 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
76imp 405 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
84, 7orim12d 962 . . . . . 6 (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥))) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
98ancoms 457 . . . . 5 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑦𝑧𝑧𝑦) → ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
109con3d 152 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦)) → ¬ (𝑦𝑧𝑧𝑦)))
11 ffvelcdm 7086 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ 𝐴)
12 ssel 3974 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝐹𝑦) ∈ 𝐴 → (𝐹𝑦) ∈ On))
1311, 12syl5 34 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝐹:ω⟶𝐴𝑦 ∈ ω) → (𝐹𝑦) ∈ On))
1413expdimp 451 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → (𝐹𝑦) ∈ On))
15 eloni 6377 . . . . . . . . 9 ((𝐹𝑦) ∈ On → Ord (𝐹𝑦))
1614, 15syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹𝑦)))
17 ffvelcdm 7086 . . . . . . . . . . 11 ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ 𝐴)
18 ssel 3974 . . . . . . . . . . 11 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐹𝑧) ∈ On))
1917, 18syl5 34 . . . . . . . . . 10 (𝐴 ⊆ On → ((𝐹:ω⟶𝐴𝑧 ∈ ω) → (𝐹𝑧) ∈ On))
2019expdimp 451 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → (𝐹𝑧) ∈ On))
21 eloni 6377 . . . . . . . . 9 ((𝐹𝑧) ∈ On → Ord (𝐹𝑧))
2220, 21syl6 35 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → Ord (𝐹𝑧)))
2316, 22anim12d 607 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧))))
2423imp 405 . . . . . 6 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹𝑦) ∧ Ord (𝐹𝑧)))
25 ordtri3 6403 . . . . . 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 713 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) ↔ ¬ ((𝐹𝑦) ∈ (𝐹𝑧) ∨ (𝐹𝑧) ∈ (𝐹𝑦))))
28 nnord 7875 . . . . . 6 (𝑦 ∈ ω → Ord 𝑦)
29 nnord 7875 . . . . . 6 (𝑧 ∈ ω → Ord 𝑧)
30 ordtri3 6403 . . . . . 6 ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3128, 29, 30syl2an 594 . . . . 5 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3231adantl 480 . . . 4 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (𝑦 = 𝑧 ↔ ¬ (𝑦𝑧𝑧𝑦)))
3310, 27, 323imtr4d 293 . . 3 ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
3433ralrimivva 3191 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
35 dff13 7261 . 2 (𝐹:ω–1-1𝐴 ↔ (𝐹:ω⟶𝐴 ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
361, 34, 35sylanbrc 581 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 205  wa 394  wo 845   = wceq 1534  wcel 2099  wral 3051  wss 3948  Ord word 6366  Oncon0 6367  suc csuc 6369  wf 6541  1-1wf1 6542  cfv 6545  ωcom 7867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3466  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3968  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-br 5146  df-opab 5208  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fv 6553  df-om 7868
This theorem is referenced by:  unblem4  9324
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