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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 6391 | . 2 ⊢ Tr On | |
2 | trpred 6336 | . 2 ⊢ ((Tr On ∧ 𝐴 ∈ On) → Pred( E , On, 𝐴) = 𝐴) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Tr wtr 5262 E cep 5577 Predcpred 6303 Oncon0 6368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-tr 5263 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 |
This theorem is referenced by: dfrecs3 8394 dfrecs3OLD 8395 tfr2ALT 8423 tfr3ALT 8424 on2recsov 8690 on2ind 8691 on3ind 8692 |
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