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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 2386. (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 10010 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑧 𝑦 ∈ 𝑥) | |
2 | 1 | aecoms 2446 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-reg 9055 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3938 df-un 3940 df-nul 4291 df-sn 4567 df-pr 4569 |
This theorem is referenced by: axrepnd 10015 |
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