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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 6116 | . . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | 1 | elin2 4124 | . 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
3 | elpred.1 | . . . 4 ⊢ 𝑌 ∈ V | |
4 | 3 | eliniseg 5925 | . . 3 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋)) |
5 | 4 | anbi2d 631 | . 2 ⊢ (𝑋 ∈ 𝐷 → ((𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) |
6 | 2, 5 | syl5bb 286 | 1 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 {csn 4525 class class class wbr 5030 ◡ccnv 5518 “ cima 5522 Predcpred 6115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 |
This theorem is referenced by: predpo 6134 setlikespec 6137 preddowncl 6143 wfrlem10 7947 preduz 13024 predfz 13027 wzel 33224 wsuclem 33225 fprlem2 33251 |
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