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 1809 through ax-9 2118, 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 2376. It is preferred to use ax6ev 1969 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 2181	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2		ax13lem1 2378	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
3		ax6ev 1969	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝑤
4		equtr 2020	. . . . . 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 2007  ax-12 2177  ax-13 2376

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

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  2703  dtrucor2  5342  axextnd  10605  ax8dfeq  35816  bj-axc10  36801  bj-alequex  36802  ax6er  36851  exlimiieq1  36852  wl-exeq  37552  wl-equsald  37557  ax6e2nd  44583  ax6e2ndVD  44932  ax6e2ndALT  44954  spd  49542