Theorem omsmo 8653

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 (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1→𝐴)

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.

Step	Hyp	Ref	Expression
1		simplr 781	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴)
2		omsmolem 8652	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧))))
3	2	adantl 487	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧))))
4	3	imp 412	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
5		omsmolem 8652	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
6	5	adantr 486	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
7	6	imp 412	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)))
8	4, 7	orim12d 979	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥))) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) → ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
9	8	ancoms 464	. . . . 5 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) → ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
10	9	con3d 153	. . . 4 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦)) → ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
11		ffvelcdm 7077	. . . . . . . . . . 11 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑦 ∈ ω) → (𝐹‘𝑦) ∈ 𝐴)
12		ssel 3925	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑦) ∈ 𝐴 → (𝐹‘𝑦) ∈ On))
13	11, 12	syl5 35	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On → ((𝐹:ω⟶𝐴 ∧ 𝑦 ∈ ω) → (𝐹‘𝑦) ∈ On))
14	13	expdimp 458	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → (𝐹‘𝑦) ∈ On))
15		eloni 6369	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) ∈ On → Ord (𝐹‘𝑦))
16	14, 15	syl6 36	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹‘𝑦)))
17		ffvelcdm 7077	. . . . . . . . . . 11 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ 𝐴)
18		ssel 3925	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐹‘𝑧) ∈ On))
19	17, 18	syl5 35	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On → ((𝐹:ω⟶𝐴 ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ On))
20	19	expdimp 458	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ On))
21		eloni 6369	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) ∈ On → Ord (𝐹‘𝑧))
22	20, 21	syl6 36	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → Ord (𝐹‘𝑧)))
23	16, 22	anim12d 621	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧))))
24	23	imp 412	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧)))
25		ordtri3 6396	. . . . . 6 ⊢ ((Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
26	24, 25	syl 18	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
27	26	adantlr 728	. . . 4 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
28		nnord 7876	. . . . . 6 ⊢ (𝑦 ∈ ω → Ord 𝑦)
29		nnord 7876	. . . . . 6 ⊢ (𝑧 ∈ ω → Ord 𝑧)
30		ordtri3 6396	. . . . . 6 ⊢ ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
31	28, 29, 30	syl2an 608	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
32	31	adantl 487	. . . 4 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
33	10, 27, 32	3imtr4d 297	. . 3 ⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
34	33	ralrimivva 3205	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
35		dff13 7254	. 2 ⊢ (𝐹:ω–1-1→𝐴 ↔ (𝐹:ω⟶𝐴 ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
36	1, 34, 35	sylanbrc 595	1 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1→𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  Ord word 6358  Oncon0 6359  suc csuc 6361  ⟶wf 6531  –1-1→wf1 6532  ‘cfv 6535  ωcom 7868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fv 6543  df-om 7869

This theorem is used by:  unblem4  9272