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| Description: The predecessor of an ordinal under E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by BJ, 16-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| predon | ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tron 6407 | . 2 ⊢ Tr On | |
| 2 | trpred 6352 | . 2 ⊢ ((Tr On ∧ 𝐴 ∈ On) → Pred( E , On, 𝐴) = 𝐴) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Tr wtr 5259 E cep 5583 Predcpred 6320 Oncon0 6384 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 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-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 | 
| This theorem is referenced by: dfrecs3 8412 dfrecs3OLD 8413 tfr2ALT 8441 tfr3ALT 8442 on2recsov 8706 on2ind 8707 on3ind 8708 | 
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