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Mirrors > Home > MPE Home > Th. List > int0OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of int0 4425 as of 26-Jul-2021. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
int0OLD | ⊢ ∩ ∅ = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3878 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
2 | 1 | pm2.21i 115 | . . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
3 | 2 | ax-gen 1713 | . . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) |
4 | equid 1926 | . . . 4 ⊢ 𝑥 = 𝑥 | |
5 | 3, 4 | 2th 253 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥) |
6 | 5 | abbii 2726 | . 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥} |
7 | df-int 4411 | . 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} | |
8 | df-v 3175 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2642 | 1 ⊢ ∩ ∅ = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 = wceq 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 ∅c0 3874 ∩ cint 4410 |
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 |
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-v 3175 df-dif 3543 df-nul 3875 df-int 4411 |
This theorem is referenced by: (None) |
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