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Mirrors > Home > MPE Home > Th. List > predonOLD | Structured version Visualization version GIF version |
Description: Obsolete version of predon 7786 as of 16-Oct-2024. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
predonOLD | ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | predep 6331 | . 2 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴)) | |
2 | onss 7785 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
3 | sseqin2 4209 | . . 3 ⊢ (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴) | |
4 | 2, 3 | sylib 217 | . 2 ⊢ (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴) |
5 | 1, 4 | eqtrd 2765 | 1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3938 ⊆ wss 3939 E cep 5575 Predcpred 6299 Oncon0 6364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 |
This theorem is referenced by: (None) |
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