![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nd4 | Structured version Visualization version GIF version |
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nd4 | ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nd3 10583 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑧 𝑦 ∈ 𝑥) | |
2 | 1 | aecoms 2427 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-13 2371 ax-ext 2703 ax-sep 5299 ax-pr 5427 ax-reg 9586 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-v 3476 df-un 3953 df-sn 4629 df-pr 4631 |
This theorem is referenced by: axrepnd 10588 |
Copyright terms: Public domain | W3C validator |