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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 9578 | . . . 4 ⊢ ¬ 𝑥 ∈ 𝑥 | |
2 | elequ2 2122 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | mtbii 326 | . . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
4 | 3 | sps 2179 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
5 | sp 2177 | . 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 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-pr 5423 ax-reg 9574 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-un 3951 df-sn 4625 df-pr 4627 |
This theorem is referenced by: nd4 10572 axrepnd 10576 axpowndlem3 10581 axinfnd 10588 axacndlem3 10591 axacnd 10594 |
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