Theorem fin1a2lem2 10361

Description: Lemma for fin1a2 10375. 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 7790	. . 3 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
3	1, 2	fmpti 7087	. 2 ⊢ 𝑆:On⟶On
4	1	fin1a2lem1 10360	. . . . . 6 ⊢ (𝑎 ∈ On → (𝑆‘𝑎) = suc 𝑎)
5	1	fin1a2lem1 10360	. . . . . 6 ⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏)
6	4, 5	eqeqan12d 2744	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ suc 𝑎 = suc 𝑏))
7		suc11 6444	. . . . 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 3178	. 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)
11		dff13 7232	. 2 ⊢ (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆‘𝑎) = (𝑆‘𝑏) → 𝑎 = 𝑏)))
12	3, 10, 11	mpbir2an 711	1 ⊢ 𝑆:On–1-1→On

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ↦ cmpt 5191  Oncon0 6335  suc csuc 6337  ⟶wf 6510  –1-1→wf1 6511  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fv 6522

This theorem is referenced by:  fin1a2lem6  10365  onsucf1o  43268