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Theorem sucexeloniOLD 7750
Description: Obsolete version of sucexeloni 7749 as of 6-Jan-2025. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sucexeloniOLD ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloniOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 onelss 6364 . . . . . . . . 9 (𝐴 ∈ On → (𝑥𝐴𝑥𝐴))
2 velsn 4607 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 eqimss 4005 . . . . . . . . . . 11 (𝑥 = 𝐴𝑥𝐴)
42, 3sylbi 216 . . . . . . . . . 10 (𝑥 ∈ {𝐴} → 𝑥𝐴)
54a1i 11 . . . . . . . . 9 (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥𝐴))
61, 5orim12d 964 . . . . . . . 8 (𝐴 ∈ On → ((𝑥𝐴𝑥 ∈ {𝐴}) → (𝑥𝐴𝑥𝐴)))
7 df-suc 6328 . . . . . . . . . 10 suc 𝐴 = (𝐴 ∪ {𝐴})
87eleq2i 2830 . . . . . . . . 9 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
9 elun 4113 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
108, 9bitr2i 276 . . . . . . . 8 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11 oridm 904 . . . . . . . 8 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
126, 10, 113imtr3g 295 . . . . . . 7 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥𝐴))
13 sssucid 6402 . . . . . . 7 𝐴 ⊆ suc 𝐴
14 sstr2 3956 . . . . . . 7 (𝑥𝐴 → (𝐴 ⊆ suc 𝐴𝑥 ⊆ suc 𝐴))
1512, 13, 14syl6mpi 67 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴))
1615ralrimiv 3143 . . . . 5 (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17 dftr3 5233 . . . . 5 (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
1816, 17sylibr 233 . . . 4 (𝐴 ∈ On → Tr suc 𝐴)
19 onss 7724 . . . . . 6 (𝐴 ∈ On → 𝐴 ⊆ On)
20 snssi 4773 . . . . . 6 (𝐴 ∈ On → {𝐴} ⊆ On)
2119, 20unssd 4151 . . . . 5 (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
227, 21eqsstrid 3997 . . . 4 (𝐴 ∈ On → suc 𝐴 ⊆ On)
23 ordon 7716 . . . . 5 Ord On
24 trssord 6339 . . . . . 6 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25243exp 1120 . . . . 5 (Tr suc 𝐴 → (suc 𝐴 ⊆ On → (Ord On → Ord suc 𝐴)))
2623, 25mpii 46 . . . 4 (Tr suc 𝐴 → (suc 𝐴 ⊆ On → Ord suc 𝐴))
2718, 22, 26sylc 65 . . 3 (𝐴 ∈ On → Ord suc 𝐴)
2827adantr 482 . 2 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → Ord suc 𝐴)
29 elong 6330 . . 3 (suc 𝐴𝑉 → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
3029adantl 483 . 2 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
3128, 30mpbird 257 1 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3065  cun 3913  wss 3915  {csn 4591  Tr wtr 5227  Ord word 6321  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by: (None)
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