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 1813 through ax-9 2118, all axioms other than ax-6 1972 are believed to be theorems of free logic, although the system without ax-6 1972 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 1974 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 2176	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2		ax13lem1 2374	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
3		ax6ev 1974	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝑤
4		equtr 2025	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑤 = 𝑦 → 𝑥 = 𝑦))
5	3, 4	eximii 1840	. . . . 5 ⊢ ∃𝑥(𝑤 = 𝑦 → 𝑥 = 𝑦)
6	5	19.35i 1882	. . . 4 ⊢ (∀𝑥 𝑤 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
7	2, 6	syl6com 37	. . 3 ⊢ (𝑤 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦))
8		ax6ev 1974	. . 3 ⊢ ∃𝑤 𝑤 = 𝑦
9	7, 8	exlimiiv 1935	. 2 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
10	1, 9	pm2.61i 182	1 ⊢ ∃𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

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  2705  dtrucor2  5290  axextnd  10278  ax8dfeq  33680  bj-axc10  34892  bj-alequex  34893  ax6er  34943  exlimiieq1  34944  wl-exeq  35620  wl-equsald  35625  ax6e2nd  42067  ax6e2ndVD  42417  ax6e2ndALT  42439  spd  46270