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Mirrors > Home > MPE Home > Th. List > elpred | Structured version Visualization version GIF version |
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) |
Ref | Expression |
---|---|
elpred.1 | ⊢ 𝑌 ∈ V |
Ref | Expression |
---|---|
elpred | ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 5680 | . . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | 1 | elin2 3801 | . 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
3 | elpred.1 | . . . 4 ⊢ 𝑌 ∈ V | |
4 | 3 | eliniseg 5494 | . . 3 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋)) |
5 | 4 | anbi2d 740 | . 2 ⊢ (𝑋 ∈ 𝐷 → ((𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) |
6 | 2, 5 | syl5bb 272 | 1 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 Vcvv 3200 {csn 4177 class class class wbr 4653 ◡ccnv 5113 “ cima 5117 Predcpred 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 |
This theorem is referenced by: predpo 5698 setlikespec 5701 preddowncl 5707 wfrlem10 7424 preduz 12461 predfz 12464 wzel 31771 wzelOLD 31772 wsuclem 31773 wsuclemOLD 31774 |
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