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Mirrors > Home > MPE Home > Th. List > predonOLD | Structured version Visualization version GIF version |
Description: Obsolete version of predon 7777 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 7776 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
3 | sseqin2 4215 | . . 3 ⊢ (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴) | |
4 | 2, 3 | sylib 217 | . 2 ⊢ (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴) |
5 | 1, 4 | eqtrd 2771 | 1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∩ cin 3947 ⊆ wss 3948 E cep 5579 Predcpred 6299 Oncon0 6364 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 |
This theorem is referenced by: (None) |
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