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| Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7778 and pwnex 7779. See the comment of abnexg 7776. (Contributed by BJ, 2-May-2021.) | 
| Ref | Expression | 
|---|---|
| abnex | ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vprc 5315 | . 2 ⊢ ¬ V ∈ V | |
| 2 | alral 3075 | . . 3 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹)) | |
| 3 | rexv 3509 | . . . . . . 7 ⊢ (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹) | |
| 4 | 3 | bicomi 224 | . . . . . 6 ⊢ (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹) | 
| 5 | 4 | abbii 2809 | . . . . 5 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} | 
| 6 | 5 | eleq1i 2832 | . . . 4 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V) | 
| 7 | 6 | biimpi 216 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V) | 
| 8 | abnexg 7776 | . . 3 ⊢ (∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V → V ∈ V)) | |
| 9 | 2, 7, 8 | syl2im 40 | . 2 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → V ∈ V)) | 
| 10 | 1, 9 | mtoi 199 | 1 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 Vcvv 3480 | 
| 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-11 2157 ax-ext 2708 ax-sep 5296 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-sn 4627 df-uni 4908 df-iun 4993 | 
| This theorem is referenced by: snnex 7778 pwnex 7779 | 
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