Theorem omsmo 8573

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 768	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴)
2		omsmolem 8572	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧))))
3	2	adantl 481	. . . . . . . 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 406	. . . . . . 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 8572	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
6	5	adantr 480	. . . . . . . 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 406	. . . . . . 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 966	. . . . . 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 458	. . . . 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		ffvelcdm 7014	. . . . . . . . . . 11 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑦 ∈ ω) → (𝐹‘𝑦) ∈ 𝐴)
12		ssel 3923	. . . . . . . . . . 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 452	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → (𝐹‘𝑦) ∈ On))
15		eloni 6316	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) ∈ On → Ord (𝐹‘𝑦))
16	14, 15	syl6 35	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹‘𝑦)))
17		ffvelcdm 7014	. . . . . . . . . . 11 ⊢ ((𝐹:ω⟶𝐴 ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ 𝐴)
18		ssel 3923	. . . . . . . . . . 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 452	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ On))
21		eloni 6316	. . . . . . . . 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 406	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧)))
25		ordtri3 6342	. . . . . 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 715	. . . 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 7804	. . . . . 6 ⊢ (𝑦 ∈ ω → Ord 𝑦)
29		nnord 7804	. . . . . 6 ⊢ (𝑧 ∈ ω → Ord 𝑧)
30		ordtri3 6342	. . . . . 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 481	. . . 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 3175	. 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 7188	. 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 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897  Ord word 6305  Oncon0 6306  suc csuc 6308  ⟶wf 6477  –1-1→wf1 6478  ‘cfv 6481  ωcom 7796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fv 6489  df-om 7797

This theorem is referenced by:  unblem4  9179