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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 9636 | . . . 4 ⊢ ¬ 𝑥 ∈ 𝑥 | |
| 2 | elequ2 2123 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦)) | |
| 3 | 1, 2 | mtbii 326 | . . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
| 4 | 3 | sps 2185 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
| 5 | sp 2183 | . 2 ⊢ (∀𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦) | |
| 6 | 4, 5 | nsyl 140 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: nd4 10630 axrepnd 10634 axpowndlem3 10639 axinfnd 10646 axacndlem3 10649 axacnd 10652 |
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