Theorem ax6e 2381

Description: At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-4 1809 through ax-9 2119, all axioms other than ax-6 1967 are believed to be theorems of free logic, although the system without ax-6 1967 is not complete in free logic.

Usage of this theorem is discouraged because it depends on ax-13 2370. It is preferred to use ax6ev 1969 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2372 became available. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6e	⊢ ∃𝑥 𝑥 = 𝑦

Proof of Theorem ax6e

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.8a 2182	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2		ax13lem1 2372	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
3		ax6ev 1969	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝑤
4		equtr 2021	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑤 = 𝑦 → 𝑥 = 𝑦))
5	3, 4	eximii 1837	. . . . 5 ⊢ ∃𝑥(𝑤 = 𝑦 → 𝑥 = 𝑦)
6	5	19.35i 1878	. . . 4 ⊢ (∀𝑥 𝑤 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
7	2, 6	syl6com 37	. . 3 ⊢ (𝑤 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦))
8		ax6ev 1969	. . 3 ⊢ ∃𝑤 𝑤 = 𝑦
9	7, 8	exlimiiv 1931	. 2 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
10	1, 9	pm2.61i 182	1 ⊢ ∃𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538  ∃wex 1779

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 2008  ax-12 2178  ax-13 2370

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  ax6  2382  spimt  2384  spim  2385  spimed  2386  spimvALT  2389  spei  2392  equs4  2414  equsal  2415  equsexALT  2417  equvini  2453  equvel  2454  2ax6elem  2468  axi9  2697  dtrucor2  5327  axextnd  10544  ax8dfeq  35786  bj-axc10  36771  bj-alequex  36772  ax6er  36821  exlimiieq1  36822  wl-exeq  37522  wl-equsald  37527  ax6e2nd  44548  ax6e2ndVD  44897  ax6e2ndALT  44919  spd  49667