![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nd2 | Structured version Visualization version GIF version |
Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.) |
Ref | Expression |
---|---|
nd2 | ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirrv 8655 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
2 | stdpc4 2497 | . . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦) | |
3 | 1 | nfnth 1874 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧 |
4 | elequ2 2157 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)) | |
5 | 3, 4 | sbie 2553 | . . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧) |
6 | 2, 5 | sylib 208 | . . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧) |
7 | 1, 6 | mto 188 | . 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦 |
8 | axc11 2464 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦)) | |
9 | 7, 8 | mtoi 190 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1627 [wsb 2047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1868 ax-4 1883 ax-5 1989 ax-6 2055 ax-7 2091 ax-9 2152 ax-10 2172 ax-11 2188 ax-12 2201 ax-13 2406 ax-ext 2749 ax-sep 4911 ax-nul 4919 ax-pr 5033 ax-reg 8651 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1632 df-ex 1851 df-nf 1856 df-sb 2048 df-clab 2756 df-cleq 2762 df-clel 2765 df-nfc 2900 df-ral 3064 df-rex 3065 df-v 3350 df-dif 3723 df-un 3725 df-nul 4061 df-sn 4314 df-pr 4316 |
This theorem is referenced by: axrepnd 9616 axpownd 9623 axinfndlem1 9627 axacndlem4 9632 |
Copyright terms: Public domain | W3C validator |