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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.) |
Ref | Expression |
---|---|
predon | ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | predep 6176 | . 2 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴)) | |
2 | onss 7507 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
3 | sseqin2 4194 | . . 3 ⊢ (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴) | |
4 | 2, 3 | sylib 220 | . 2 ⊢ (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴) |
5 | 1, 4 | eqtrd 2858 | 1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 E cep 5466 Predcpred 6149 Oncon0 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 |
This theorem is referenced by: dfrecs3 8011 tfr2ALT 8039 tfr3ALT 8040 |
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