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Mirrors > Home > MPE Home > Th. List > zfregOLD | Structured version Visualization version GIF version |
Description: Obsolete version of zfreg 8383 as of 28-Apr-2021. (Contributed by NM, 26-Nov-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
zfregOLD.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
zfregOLD | ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfregOLD.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | zfregclOLD 8384 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴) |
3 | n0 3890 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
4 | disj 3969 | . . 3 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴) | |
5 | 4 | rexbii 3023 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴) |
6 | 2, 3, 5 | 3imtr4i 280 | 1 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-in 3547 df-nul 3875 |
This theorem is referenced by: (None) |
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