Theorem ax6e 2383

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 1810 through ax-9 2121, all axioms other than ax-6 1968 are believed to be theorems of free logic, although the system without ax-6 1968 is not complete in free logic.

Usage of this theorem is discouraged because it depends on ax-13 2372. It is preferred to use ax6ev 1970 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2374 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 2184	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2		ax13lem1 2374	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
3		ax6ev 1970	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝑤
4		equtr 2022	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑤 = 𝑦 → 𝑥 = 𝑦))
5	3, 4	eximii 1838	. . . . 5 ⊢ ∃𝑥(𝑤 = 𝑦 → 𝑥 = 𝑦)
6	5	19.35i 1879	. . . 4 ⊢ (∀𝑥 𝑤 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
7	2, 6	syl6com 37	. . 3 ⊢ (𝑤 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦))
8		ax6ev 1970	. . 3 ⊢ ∃𝑤 𝑤 = 𝑦
9	7, 8	exlimiiv 1932	. 2 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
10	1, 9	pm2.61i 182	1 ⊢ ∃𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  ax6  2384  spimt  2386  spim  2387  spimed  2388  spimvALT  2391  spei  2394  equs4  2416  equsal  2417  equsexALT  2419  equvini  2455  equvel  2456  2ax6elem  2470  axi9  2699  dtrucor2  5310  axextnd  10479  ax8dfeq  35831  bj-axc10  36816  bj-alequex  36817  ax6er  36866  exlimiieq1  36867  wl-exeq  37567  wl-equsald  37572  ax6e2nd  44590  ax6e2ndVD  44939  ax6e2ndALT  44961  spd  49709