Theorem omsmo 8488

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 766	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴)
2		omsmolem 8487	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧))))
3	2	adantl 482	. . . . . . . 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 407	. . . . . . 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 8487	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
6	5	adantr 481	. . . . . . . 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 407	. . . . . . 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 962	. . . . . 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 459	. . . . 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 152	. . . 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		ffvelrn 6959	. . . . . . . . . . 11 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑦 ∈ ω) → (𝐹‘𝑦) ∈ 𝐴)
12		ssel 3914	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑦) ∈ 𝐴 → (𝐹‘𝑦) ∈ On))
13	11, 12	syl5 34	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On → ((𝐹:ω⟶𝐴 ∧ 𝑦 ∈ ω) → (𝐹‘𝑦) ∈ On))
14	13	expdimp 453	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → (𝐹‘𝑦) ∈ On))
15		eloni 6276	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) ∈ On → Ord (𝐹‘𝑦))
16	14, 15	syl6 35	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹‘𝑦)))
17		ffvelrn 6959	. . . . . . . . . . 11 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ 𝐴)
18		ssel 3914	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐹‘𝑧) ∈ On))
19	17, 18	syl5 34	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On → ((𝐹:ω⟶𝐴 ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ On))
20	19	expdimp 453	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ On))
21		eloni 6276	. . . . . . . . 9 ⊢ ((𝐹‘𝑧) ∈ On → Ord (𝐹‘𝑧))
22	20, 21	syl6 35	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → Ord (𝐹‘𝑧)))
23	16, 22	anim12d 609	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧))))
24	23	imp 407	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧)))
25		ordtri3 6302	. . . . . 6 ⊢ ((Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
26	24, 25	syl 17	. . . . 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 712	. . . 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 7720	. . . . . 6 ⊢ (𝑦 ∈ ω → Ord 𝑦)
29		nnord 7720	. . . . . 6 ⊢ (𝑧 ∈ ω → Ord 𝑧)
30		ordtri3 6302	. . . . . 6 ⊢ ((Ord 𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
31	28, 29, 30	syl2an 596	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦)))
32	31	adantl 482	. . . 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 294	. . 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 3123	. 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 7128	. 2 ⊢ (𝐹:ω–1-1→𝐴 ↔ (𝐹:ω⟶𝐴 ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
36	1, 34, 35	sylanbrc 583	1 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  Ord word 6265  Oncon0 6266  suc csuc 6268  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433  ωcom 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441  df-om 7713

This theorem is referenced by:  unblem4  9069