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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 8504 | . . . 4 ⊢ ¬ 𝑥 ∈ 𝑥 | |
2 | elequ2 2004 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | mtbii 316 | . . 3 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
4 | 3 | sps 2055 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦) |
5 | sp 2053 | . 2 ⊢ (∀𝑧 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑦) | |
6 | 4, 5 | nsyl 135 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 |
This theorem is referenced by: nd4 9412 axrepnd 9416 axpowndlem3 9421 axinfnd 9428 axacndlem3 9431 axacnd 9434 |
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