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Theorem sucexeloniOLD 7745
Description: Obsolete version of sucexeloni 7744 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 6359 . . . . . . . . 9 (𝐴 ∈ On → (𝑥𝐴𝑥𝐴))
2 velsn 4602 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 eqimss 4000 . . . . . . . . . . 11 (𝑥 = 𝐴𝑥𝐴)
42, 3sylbi 216 . . . . . . . . . 10 (𝑥 ∈ {𝐴} → 𝑥𝐴)
54a1i 11 . . . . . . . . 9 (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥𝐴))
61, 5orim12d 963 . . . . . . . 8 (𝐴 ∈ On → ((𝑥𝐴𝑥 ∈ {𝐴}) → (𝑥𝐴𝑥𝐴)))
7 df-suc 6323 . . . . . . . . . 10 suc 𝐴 = (𝐴 ∪ {𝐴})
87eleq2i 2829 . . . . . . . . 9 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
9 elun 4108 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
108, 9bitr2i 275 . . . . . . . 8 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11 oridm 903 . . . . . . . 8 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
126, 10, 113imtr3g 294 . . . . . . 7 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥𝐴))
13 sssucid 6397 . . . . . . 7 𝐴 ⊆ suc 𝐴
14 sstr2 3951 . . . . . . 7 (𝑥𝐴 → (𝐴 ⊆ suc 𝐴𝑥 ⊆ suc 𝐴))
1512, 13, 14syl6mpi 67 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴))
1615ralrimiv 3142 . . . . 5 (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17 dftr3 5228 . . . . 5 (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
1816, 17sylibr 233 . . . 4 (𝐴 ∈ On → Tr suc 𝐴)
19 onss 7719 . . . . . 6 (𝐴 ∈ On → 𝐴 ⊆ On)
20 snssi 4768 . . . . . 6 (𝐴 ∈ On → {𝐴} ⊆ On)
2119, 20unssd 4146 . . . . 5 (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
227, 21eqsstrid 3992 . . . 4 (𝐴 ∈ On → suc 𝐴 ⊆ On)
23 ordon 7711 . . . . 5 Ord On
24 trssord 6334 . . . . . 6 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25243exp 1119 . . . . 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 481 . 2 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → Ord suc 𝐴)
29 elong 6325 . . 3 (suc 𝐴𝑉 → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
3029adantl 482 . 2 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
3128, 30mpbird 256 1 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3064  cun 3908  wss 3910  {csn 4586  Tr wtr 5222  Ord word 6316  Oncon0 6317  suc csuc 6319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-tr 5223  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-ord 6320  df-on 6321  df-suc 6323
This theorem is referenced by: (None)
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