Theorem fin1a2lem2 10470

Description: Lemma for fin1a2 10484. The successor operation on the ordinal numbers is injective or one-to-one. Lemma 1.17 of [Schloeder] p. 2. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Hypothesis

Ref	Expression
fin1a2lem.a	⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)

Assertion

Ref	Expression
fin1a2lem2	⊢ 𝑆:On–1-1→On

Proof of Theorem fin1a2lem2

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin1a2lem.a	. . 3 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
2		onsuc 7847	. . 3 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
3	1, 2	fmpti 7146	. 2 ⊢ 𝑆:On⟶On
4	1	fin1a2lem1 10469	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑆‘𝑎) = suc 𝑎)
5	1	fin1a2lem1 10469	. . . . . 6 ⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏)
6	4, 5	eqeqan12d 2754	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ suc 𝑎 = suc 𝑏))
7		suc11 6502	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏))
8	6, 7	bitrd 279	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ 𝑎 = 𝑏))
9	8	biimpd 229	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏))
10	9	rgen2 3205	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)
11		dff13 7292	. 2 ⊢ (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)))
12	3, 10, 11	mpbir2an 710	1 ⊢ 𝑆:On–1-1→On

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ↦ cmpt 5249  Oncon0 6395  suc csuc 6397  ⟶wf 6569  –1-1→wf1 6570  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fv 6581

This theorem is referenced by:  fin1a2lem6  10474  onsucf1o  43234