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Mirrors > Home > ILE Home > Th. List > csbprc | 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.) |
Ref | Expression |
---|---|
csbprc | ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3050 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | sbcex 2963 | . . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | con3i 627 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵) |
4 | 3 | pm2.21d 614 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → ⊥)) |
5 | falim 1362 | . . . . 5 ⊢ (⊥ → [𝐴 / 𝑥]𝑦 ∈ 𝐵) | |
6 | 4, 5 | impbid1 141 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥)) |
7 | 6 | abbidv 2288 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥}) |
8 | fal 1355 | . . . 4 ⊢ ¬ ⊥ | |
9 | 8 | abf 3458 | . . 3 ⊢ {𝑦 ∣ ⊥} = ∅ |
10 | 7, 9 | eqtrdi 2219 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = ∅) |
11 | 1, 10 | eqtrid 2215 | 1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 ⊥wfal 1353 ∈ wcel 2141 {cab 2156 Vcvv 2730 [wsbc 2955 ⦋csb 3049 ∅c0 3414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 |
This theorem is referenced by: (None) |
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