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Mirrors > Home > MPE Home > Th. List > predprc | Structured version Visualization version GIF version |
Description: The predecessor of a proper class is empty. (Contributed by Scott Fenton, 25-Nov-2024.) |
Ref | Expression |
---|---|
predprc | ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6308 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | snprc 4724 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ V ↔ {𝑋} = ∅) | |
3 | 2 | biimpi 215 | . . . . . 6 ⊢ (¬ 𝑋 ∈ V → {𝑋} = ∅) |
4 | 3 | imaeq2d 6066 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (◡𝑅 “ {𝑋}) = (◡𝑅 “ ∅)) |
5 | ima0 6083 | . . . . 5 ⊢ (◡𝑅 “ ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2783 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (◡𝑅 “ {𝑋}) = ∅) |
7 | 6 | ineq2d 4212 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ ∅)) |
8 | in0 4393 | . . 3 ⊢ (𝐴 ∩ ∅) = ∅ | |
9 | 7, 8 | eqtrdi 2783 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = ∅) |
10 | 1, 9 | eqtrid 2779 | 1 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ∩ cin 3946 ∅c0 4324 {csn 4630 ◡ccnv 5679 “ cima 5683 Predcpred 6307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5151 df-opab 5213 df-xp 5686 df-cnv 5688 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 |
This theorem is referenced by: predres 6348 |
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