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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 2956 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | sbcex 2870 | . . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | con3i 602 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵) |
4 | 3 | pm2.21d 589 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → ⊥)) |
5 | falim 1313 | . . . . 5 ⊢ (⊥ → [𝐴 / 𝑥]𝑦 ∈ 𝐵) | |
6 | 4, 5 | impbid1 141 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥)) |
7 | 6 | abbidv 2217 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥}) |
8 | fal 1306 | . . . 4 ⊢ ¬ ⊥ | |
9 | 8 | abf 3353 | . . 3 ⊢ {𝑦 ∣ ⊥} = ∅ |
10 | 7, 9 | syl6eq 2148 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = ∅) |
11 | 1, 10 | syl5eq 2144 | 1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1299 ⊥wfal 1304 ∈ wcel 1448 {cab 2086 Vcvv 2641 [wsbc 2862 ⦋csb 2955 ∅c0 3310 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-in 3027 df-ss 3034 df-nul 3311 |
This theorem is referenced by: (None) |
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