Theorem ax6e 2385

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 2123, 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 2374. It is preferred to use ax6ev 1970 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2376 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 2186	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2		ax13lem1 2376	. . . 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 2182  ax-13 2374

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

This theorem is referenced by:  ax6  2386  spimt  2388  spim  2389  spimed  2390  spimvALT  2393  spei  2396  equs4  2418  equsal  2419  equsexALT  2421  equvini  2457  equvel  2458  2ax6elem  2472  axi9  2701  dtrucor2  5312  axextnd  10489  ax8dfeq  35861  bj-axc10  36848  bj-alequex  36849  ax6er  36898  exlimiieq1  36899  wl-exeq  37599  wl-equsald  37604  ax6e2nd  44675  ax6e2ndVD  45024  ax6e2ndALT  45046  spd  49803