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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 6150 | . . . . 5 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | 1 | elin2 4176 | . . . 4 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
3 | 2 | baib 538 | . . 3 ⊢ (𝑌 ∈ 𝐴 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
4 | 3 | adantl 484 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
5 | elimasng 5957 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 〈𝑋, 𝑌〉 ∈ ◡𝑅)) | |
6 | df-br 5069 | . . 3 ⊢ (𝑋◡𝑅𝑌 ↔ 〈𝑋, 𝑌〉 ∈ ◡𝑅) | |
7 | 5, 6 | syl6bbr 291 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 𝑋◡𝑅𝑌)) |
8 | brcnvg 5752 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑋◡𝑅𝑌 ↔ 𝑌𝑅𝑋)) | |
9 | 4, 7, 8 | 3bitrd 307 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 {csn 4569 〈cop 4575 class class class wbr 5068 ◡ccnv 5556 “ cima 5560 Predcpred 6149 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 |
This theorem is referenced by: predpo 6168 predpoirr 6178 predfrirr 6179 wfrlem10 7966 wsuclem 33114 wsuclb 33117 |
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