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Mirrors > Home > MPE Home > Th. List > nd2 | Structured version Visualization version GIF version |
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nd2 | ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirrv 9342 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
2 | stdpc4 2071 | . . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦) | |
3 | 1 | nfnth 1805 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧 |
4 | elequ2 2121 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)) | |
5 | 3, 4 | sbie 2506 | . . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧) |
6 | 2, 5 | sylib 217 | . . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧) |
7 | 1, 6 | mto 196 | . 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦 |
8 | axc11 2430 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦)) | |
9 | 7, 8 | mtoi 198 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 [wsb 2067 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-13 2372 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 ax-reg 9338 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-v 3431 df-dif 3889 df-un 3891 df-nul 4257 df-sn 4562 df-pr 4564 |
This theorem is referenced by: axrepnd 10360 axpownd 10367 axinfndlem1 10371 axacndlem4 10376 |
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