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Mirrors > Home > MPE Home > Th. List > elpredg | Structured version Visualization version GIF version |
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) |
Ref | Expression |
---|---|
elpredg | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 5983 | . . . . 5 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | 1 | elin2 4056 | . . . 4 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
3 | 2 | baib 528 | . . 3 ⊢ (𝑌 ∈ 𝐴 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
4 | 3 | adantl 474 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
5 | elimasng 5792 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 〈𝑋, 𝑌〉 ∈ ◡𝑅)) | |
6 | df-br 4926 | . . 3 ⊢ (𝑋◡𝑅𝑌 ↔ 〈𝑋, 𝑌〉 ∈ ◡𝑅) | |
7 | 5, 6 | syl6bbr 281 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 𝑋◡𝑅𝑌)) |
8 | brcnvg 5596 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑋◡𝑅𝑌 ↔ 𝑌𝑅𝑋)) | |
9 | 4, 7, 8 | 3bitrd 297 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∈ wcel 2050 {csn 4435 〈cop 4441 class class class wbr 4925 ◡ccnv 5402 “ cima 5406 Predcpred 5982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-opab 4988 df-xp 5409 df-cnv 5411 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 |
This theorem is referenced by: predpo 6001 predpoirr 6011 predfrirr 6012 wfrlem10 7766 wsuclem 32662 wsuclb 32665 |
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