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Mirrors > Home > MPE Home > Th. List > nd1 | Structured version Visualization version GIF version |
Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.) |
Ref | Expression |
---|---|
nd1 | ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirrv 8387 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
2 | stdpc4 2341 | . . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → [𝑧 / 𝑦]𝑦 ∈ 𝑧) | |
3 | 1 | nfnth 1719 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧 |
4 | elequ1 1984 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧)) | |
5 | 3, 4 | sbie 2396 | . . . 4 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧) |
6 | 2, 5 | sylib 207 | . . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → 𝑧 ∈ 𝑧) |
7 | 1, 6 | mto 187 | . 2 ⊢ ¬ ∀𝑦 𝑦 ∈ 𝑧 |
8 | axc11 2302 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦 ∈ 𝑧 → ∀𝑦 𝑦 ∈ 𝑧)) | |
9 | 7, 8 | mtoi 189 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 |
This theorem is referenced by: axrepnd 9295 axinfndlem1 9306 axinfnd 9307 axacndlem1 9308 axacndlem2 9309 |
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