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Theorem fin1a2lem2 10438
Description: Lemma for fin1a2 10452. 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.
StepHypRef Expression
1 fin1a2lem.a . . 3 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
2 onsuc 7830 . . 3 (𝑥 ∈ On → suc 𝑥 ∈ On)
31, 2fmpti 7131 . 2 𝑆:On⟶On
41fin1a2lem1 10437 . . . . . 6 (𝑎 ∈ On → (𝑆𝑎) = suc 𝑎)
51fin1a2lem1 10437 . . . . . 6 (𝑏 ∈ On → (𝑆𝑏) = suc 𝑏)
64, 5eqeqan12d 2748 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) ↔ suc 𝑎 = suc 𝑏))
7 suc11 6492 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏𝑎 = 𝑏))
86, 7bitrd 279 . . . 4 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) ↔ 𝑎 = 𝑏))
98biimpd 229 . . 3 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏))
109rgen2 3196 . 2 𝑎 ∈ On ∀𝑏 ∈ On ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏)
11 dff13 7274 . 2 (𝑆:On–1-1→On ↔ (𝑆:On⟶On ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝑆𝑎) = (𝑆𝑏) → 𝑎 = 𝑏)))
123, 10, 11mpbir2an 711 1 𝑆:On–1-1→On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  cmpt 5230  Oncon0 6385  suc csuc 6387  wf 6558  1-1wf1 6559  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fv 6570
This theorem is referenced by:  fin1a2lem6  10442  onsucf1o  43261
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