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Mirrors > Home > MPE Home > Th. List > nd3 | Structured version Visualization version GIF version |
Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.) |
Ref | Expression |
---|---|
nd3 | ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirrv 9060 | . . . 4 ⊢ ¬ 𝑥 ∈ 𝑥 | |
2 | elequ2 2129 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | mtbii 328 | . . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
4 | 3 | sps 2184 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
5 | sp 2182 | . 2 ⊢ (∀𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦) | |
6 | 4, 5 | nsyl 142 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-reg 9056 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3939 df-un 3941 df-nul 4292 df-sn 4568 df-pr 4570 |
This theorem is referenced by: nd4 10012 axrepnd 10016 axpowndlem3 10021 axinfnd 10028 axacndlem3 10031 axacnd 10034 |
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