Theorem ax6e 2391

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 1807 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 2380. It is preferred to use ax6ev 1969 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2382 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 2182	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2		ax13lem1 2382	. . . 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 1835	. . . . 5 ⊢ ∃𝑥(𝑤 = 𝑦 → 𝑥 = 𝑦)
6	5	19.35i 1877	. . . 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 1930	. 2 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
10	1, 9	pm2.61i 182	1 ⊢ ∃𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778

This theorem is referenced by:  ax6  2392  spimt  2394  spim  2395  spimed  2396  spimvALT  2399  spei  2402  equs4  2424  equsal  2425  equsexALT  2427  equvini  2463  equvel  2464  2ax6elem  2478  axi9  2707  dtrucor2  5390  axextnd  10660  ax8dfeq  35762  bj-axc10  36749  bj-alequex  36750  ax6er  36799  exlimiieq1  36800  wl-exeq  37488  wl-equsald  37493  ax6e2nd  44529  ax6e2ndVD  44879  ax6e2ndALT  44901  spd  48770