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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 2386. (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 9054 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
2 | stdpc4 2069 | . . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦) | |
3 | 1 | nfnth 1799 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧 |
4 | elequ2 2125 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)) | |
5 | 3, 4 | sbie 2540 | . . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧) |
6 | 2, 5 | sylib 220 | . . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧) |
7 | 1, 6 | mto 199 | . 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦 |
8 | axc11 2448 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦)) | |
9 | 7, 8 | mtoi 201 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 [wsb 2065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-reg 9050 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3938 df-un 3940 df-nul 4291 df-sn 4561 df-pr 4563 |
This theorem is referenced by: axrepnd 10010 axpownd 10017 axinfndlem1 10021 axacndlem4 10026 |
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