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| Mirrors > Home > MPE Home > Th. List > csbprc | Structured version Visualization version GIF version | ||
| Description: The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.) (Proof shortened by JJ, 27-Aug-2021.) |
| Ref | Expression |
|---|---|
| csbprc | ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcex 3798 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V) | |
| 2 | falim 1557 | . . . 4 ⊢ (⊥ → 𝐴 ∈ V) | |
| 3 | 1, 2 | pm5.21ni 377 | . . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥)) |
| 4 | 3 | abbidv 2808 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥}) |
| 5 | df-csb 3900 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
| 6 | fal 1554 | . . . 4 ⊢ ¬ ⊥ | |
| 7 | 6 | abf 4406 | . . 3 ⊢ {𝑦 ∣ ⊥} = ∅ |
| 8 | 7 | eqcomi 2746 | . 2 ⊢ ∅ = {𝑦 ∣ ⊥} |
| 9 | 4, 5, 8 | 3eqtr4g 2802 | 1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ⊥wfal 1552 ∈ wcel 2108 {cab 2714 Vcvv 3480 [wsbc 3788 ⦋csb 3899 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: csb0 4410 sbcel12 4411 sbcne12 4415 sbcel2 4418 csbidm 4433 csbun 4441 csbin 4442 csbdif 4524 csbif 4583 csbuni 4936 sbcbr123 5197 sbcbr 5198 csbexg 5310 csbopab 5560 csbxp 5785 csbres 6000 csbima12 6097 csbrn 6223 csbiota 6554 csbfv12 6954 csbfv 6956 csbriota 7403 csbov123 7475 csbov 7476 |
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