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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 8777 | . . . 4 ⊢ ¬ 𝑥 ∈ 𝑥 | |
2 | elequ2 2178 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | mtbii 318 | . . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
4 | 3 | sps 2226 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
5 | sp 2224 | . 2 ⊢ (∀𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦) | |
6 | 4, 5 | nsyl 138 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-reg 8773 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-v 3416 df-dif 3801 df-un 3803 df-nul 4147 df-sn 4400 df-pr 4402 |
This theorem is referenced by: nd4 9734 axrepnd 9738 axpowndlem3 9743 axinfnd 9750 axacndlem3 9753 axacnd 9756 |
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