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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 2976 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | sbcex 2890 | . . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | con3i 606 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵) |
4 | 3 | pm2.21d 593 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → ⊥)) |
5 | falim 1330 | . . . . 5 ⊢ (⊥ → [𝐴 / 𝑥]𝑦 ∈ 𝐵) | |
6 | 4, 5 | impbid1 141 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥)) |
7 | 6 | abbidv 2235 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥}) |
8 | fal 1323 | . . . 4 ⊢ ¬ ⊥ | |
9 | 8 | abf 3376 | . . 3 ⊢ {𝑦 ∣ ⊥} = ∅ |
10 | 7, 9 | syl6eq 2166 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = ∅) |
11 | 1, 10 | syl5eq 2162 | 1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1316 ⊥wfal 1321 ∈ wcel 1465 {cab 2103 Vcvv 2660 [wsbc 2882 ⦋csb 2975 ∅c0 3333 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-in 3047 df-ss 3054 df-nul 3334 |
This theorem is referenced by: (None) |
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