Theorem nulmoOLD 2758

Description: Obsolete version of nulmo 2757 as of 26-Apr-2023. (Contributed by NM, 22-Dec-2007.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nulmoOLD	⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem nulmoOLD

Step	Hyp	Ref	Expression
1		nfv 1873	. . 3 ⊢ Ⅎ𝑥⊥
2	1	axextmo 2756	. 2 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
3		nbfal 1522	. . . . 5 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥))
4	3	bicomi 216	. . . 4 ⊢ ((𝑦 ∈ 𝑥 ↔ ⊥) ↔ ¬ 𝑦 ∈ 𝑥)
5	4	albii 1782	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
6	5	mobii 2559	. 2 ⊢ (∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) ↔ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
7	2, 6	mpbi 222	1 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   ↔ wb 198  ∀wal 1505  ⊥wfal 1519  ∃*wmo 2545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547

This theorem is referenced by: (None)