Theorem ax6e 2387

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 1811 through ax-9 2124, all axioms other than ax-6 1969 are believed to be theorems of free logic, although the system without ax-6 1969 is not complete in free logic.

Usage of this theorem is discouraged because it depends on ax-13 2376. It is preferred to use ax6ev 1971 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2378 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 2189	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2		ax13lem1 2378	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
3		ax6ev 1971	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝑤
4		equtr 2023	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑤 = 𝑦 → 𝑥 = 𝑦))
5	3, 4	eximii 1839	. . . . 5 ⊢ ∃𝑥(𝑤 = 𝑦 → 𝑥 = 𝑦)
6	5	19.35i 1880	. . . 4 ⊢ (∀𝑥 𝑤 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
7	2, 6	syl6com 37	. . 3 ⊢ (𝑤 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦))
8		ax6ev 1971	. . 3 ⊢ ∃𝑤 𝑤 = 𝑦
9	7, 8	exlimiiv 1933	. 2 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
10	1, 9	pm2.61i 182	1 ⊢ ∃𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185  ax-13 2376

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  ax6  2388  spimt  2390  spim  2391  spimed  2392  spimvALT  2395  spei  2398  equs4  2420  equsal  2421  equsexALT  2423  equvini  2459  equvel  2460  2ax6elem  2474  axi9  2704  dtrucor2  5314  axextnd  10514  ax8dfeq  35978  bj-axc10  37090  bj-alequex  37091  ax6er  37140  exlimiieq1  37141  wl-exeq  37859  wl-equsald  37864  ax6e2nd  44985  ax6e2ndVD  45334  ax6e2ndALT  45356  spd  50153