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Theorem onsucf1lem 43282
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 2778 . . . . . 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 6491 . . . . . . 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 3146 . . 3 (𝐴 ∈ On → ∀𝑏 ∈ On (𝐴 = suc 𝑏𝑏 = 𝐴))
14 eqeq2 2749 . . . . 5 (𝑐 = 𝐴 → (𝑏 = 𝑐𝑏 = 𝐴))
1514imbi2d 340 . . . 4 (𝑐 = 𝐴 → ((𝐴 = suc 𝑏𝑏 = 𝑐) ↔ (𝐴 = suc 𝑏𝑏 = 𝐴)))
1615ralbidv 3178 . . 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 1914 . . 3 𝑐 𝐴 = suc 𝑏
1918rmo2 3887 . 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 1540  wex 1779  wcel 2108  wral 3061  ∃*wrmo 3379   cuni 4907  Oncon0 6384  suc csuc 6386
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  onsucf1olem  43283
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