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Mirrors > Home > MPE Home > Th. List > axregndlem1 | Structured version Visualization version GIF version |
Description: Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axregndlem1 | ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2180 | . 2 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥 𝑥 ∈ 𝑦) | |
2 | nfae 2455 | . . 3 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑧 | |
3 | nfae 2455 | . . . . . 6 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑧 | |
4 | elirrv 9062 | . . . . . . . . 9 ⊢ ¬ 𝑥 ∈ 𝑥 | |
5 | elequ1 2121 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)) | |
6 | 4, 5 | mtbii 328 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → ¬ 𝑧 ∈ 𝑥) |
7 | 6 | sps 2184 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ 𝑧 ∈ 𝑥) |
8 | 7 | pm2.21d 121 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)) |
9 | 3, 8 | alrimi 2213 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)) |
10 | 9 | anim2i 618 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑥 = 𝑧) → (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) |
11 | 10 | expcom 416 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))) |
12 | 2, 11 | eximd 2216 | . 2 ⊢ (∀𝑥 𝑥 = 𝑧 → (∃𝑥 𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))) |
13 | 1, 12 | syl5 34 | 1 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-reg 9058 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-dif 3941 df-un 3943 df-nul 4294 df-sn 4570 df-pr 4572 |
This theorem is referenced by: axregndlem2 10027 axregnd 10028 |
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