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Mirrors > Home > MPE Home > Th. List > predon | Structured version Visualization version GIF version |
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 6345 | . 2 ⊢ Tr On | |
2 | trpred 6290 | . 2 ⊢ ((Tr On ∧ 𝐴 ∈ On) → Pred( E , On, 𝐴) = 𝐴) | |
3 | 1, 2 | mpan 689 | 1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Tr wtr 5227 E cep 5541 Predcpred 6257 Oncon0 6322 |
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-10 2138 ax-11 2155 ax-12 2172 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-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 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-nul 4288 df-if 4492 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-xp 5644 df-rel 5645 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 |
This theorem is referenced by: dfrecs3 8323 dfrecs3OLD 8324 tfr2ALT 8352 tfr3ALT 8353 on2recsov 8619 on2ind 8620 on3ind 8621 |
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