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 1805 through ax-9 2115, all axioms other than ax-6 1964 are believed to be theorems of free logic, although the system without ax-6 1964 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 1966 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 2178	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
2		ax13lem1 2376	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
3		ax6ev 1966	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝑤
4		equtr 2017	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑤 = 𝑦 → 𝑥 = 𝑦))
5	3, 4	eximii 1833	. . . . 5 ⊢ ∃𝑥(𝑤 = 𝑦 → 𝑥 = 𝑦)
6	5	19.35i 1875	. . . 4 ⊢ (∀𝑥 𝑤 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
7	2, 6	syl6com 37	. . 3 ⊢ (𝑤 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦))
8		ax6ev 1966	. . 3 ⊢ ∃𝑤 𝑤 = 𝑦
9	7, 8	exlimiiv 1928	. 2 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
10	1, 9	pm2.61i 182	1 ⊢ ∃𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1534  ∃wex 1775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-12 2174  ax-13 2374

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776

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  5377  axextnd  10628  ax8dfeq  35779  bj-axc10  36765  bj-alequex  36766  ax6er  36815  exlimiieq1  36816  wl-exeq  37514  wl-equsald  37519  ax6e2nd  44555  ax6e2ndVD  44905  ax6e2ndALT  44927  spd  48908