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Mirrors > Home > MPE Home > Th. List > ax6dgen | Structured version Visualization version GIF version |
Description: Tarski's system uses the weaker ax6v 1964 instead of the bundled ax-6 1963, so here we show that the degenerate case of ax-6 1963 can be derived. Even though ax-6 1963 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1963 is ax6v 1964. We later rederive from ax6v 1964 the bundled form as ax6 2377 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.) |
Ref | Expression |
---|---|
ax6dgen | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2007 | . 2 ⊢ 𝑥 = 𝑥 | |
2 | 1 | notnoti 143 | . . 3 ⊢ ¬ ¬ 𝑥 = 𝑥 |
3 | 2 | spfalw 1993 | . 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥) |
4 | 1, 3 | mt2 199 | 1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 |
This theorem depends on definitions: df-bi 206 df-ex 1774 |
This theorem is referenced by: (None) |
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