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Theorem onsucf1lem 43259
Description: For ordinals, the successor operation is injective, so there is at most one ordinal that a given ordinal could be the successor of. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
Assertion
Ref Expression
onsucf1lem (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
Distinct variable group:   𝐴,𝑏

Proof of Theorem onsucf1lem
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 onuni 7808 . . 3 (𝐴 ∈ On → 𝐴 ∈ On)
2 onsucuni2 7854 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐴 = suc 𝑏) → suc 𝐴 = 𝐴)
32adantlr 715 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → suc 𝐴 = 𝐴)
4 simpr 484 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → 𝐴 = suc 𝑏)
53, 4eqtr2d 2776 . . . . . 6 (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → suc 𝑏 = suc 𝐴)
61anim1i 615 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → ( 𝐴 ∈ On ∧ 𝑏 ∈ On))
76adantr 480 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → ( 𝐴 ∈ On ∧ 𝑏 ∈ On))
87ancomd 461 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → (𝑏 ∈ On ∧ 𝐴 ∈ On))
9 suc11 6493 . . . . . . 7 ((𝑏 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑏 = suc 𝐴𝑏 = 𝐴))
108, 9syl 17 . . . . . 6 (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → (suc 𝑏 = suc 𝐴𝑏 = 𝐴))
115, 10mpbid 232 . . . . 5 (((𝐴 ∈ On ∧ 𝑏 ∈ On) ∧ 𝐴 = suc 𝑏) → 𝑏 = 𝐴)
1211ex 412 . . . 4 ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 = suc 𝑏𝑏 = 𝐴))
1312ralrimiva 3144 . . 3 (𝐴 ∈ On → ∀𝑏 ∈ On (𝐴 = suc 𝑏𝑏 = 𝐴))
14 eqeq2 2747 . . . . 5 (𝑐 = 𝐴 → (𝑏 = 𝑐𝑏 = 𝐴))
1514imbi2d 340 . . . 4 (𝑐 = 𝐴 → ((𝐴 = suc 𝑏𝑏 = 𝑐) ↔ (𝐴 = suc 𝑏𝑏 = 𝐴)))
1615ralbidv 3176 . . 3 (𝑐 = 𝐴 → (∀𝑏 ∈ On (𝐴 = suc 𝑏𝑏 = 𝑐) ↔ ∀𝑏 ∈ On (𝐴 = suc 𝑏𝑏 = 𝐴)))
171, 13, 16spcedv 3598 . 2 (𝐴 ∈ On → ∃𝑐𝑏 ∈ On (𝐴 = suc 𝑏𝑏 = 𝑐))
18 nfv 1912 . . 3 𝑐 𝐴 = suc 𝑏
1918rmo2 3896 . 2 (∃*𝑏 ∈ On 𝐴 = suc 𝑏 ↔ ∃𝑐𝑏 ∈ On (𝐴 = suc 𝑏𝑏 = 𝑐))
2017, 19sylibr 234 1 (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wral 3059  ∃*wrmo 3377   cuni 4912  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  onsucf1olem  43260
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