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Mirrors > Home > MPE Home > Th. List > ax6e | Structured version Visualization version GIF version |
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 1812 through ax-9 2117,
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.) |
Ref | Expression |
---|---|
ax6e | ⊢ ∃𝑥 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2175 | . 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 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
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 2700 dtrucor2 5366 axextnd 10573 ax8dfeq 34701 bj-axc10 35566 bj-alequex 35567 ax6er 35616 exlimiieq1 35617 wl-exeq 36308 wl-equsald 36313 ax6e2nd 43190 ax6e2ndVD 43540 ax6e2ndALT 43562 spd 47563 |
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