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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 9063 | . . . 4 ⊢ ¬ 𝑥 ∈ 𝑥 | |
2 | elequ2 2128 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | mtbii 328 | . . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
4 | 3 | sps 2183 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
5 | sp 2181 | . 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 1534 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-reg 9059 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-dif 3942 df-un 3944 df-nul 4295 df-sn 4571 df-pr 4573 |
This theorem is referenced by: nd4 10015 axrepnd 10019 axpowndlem3 10024 axinfnd 10031 axacndlem3 10034 axacnd 10037 |
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