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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. (Contributed by NM, 3-Jan-2002.) |
Ref | Expression |
---|---|
axregndlem1 | ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2199 | . 2 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥 𝑥 ∈ 𝑦) | |
2 | nfae 2458 | . . 3 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑧 | |
3 | nfae 2458 | . . . . . 6 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑧 | |
4 | elirrv 8668 | . . . . . . . . 9 ⊢ ¬ 𝑥 ∈ 𝑥 | |
5 | elequ1 2146 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)) | |
6 | 4, 5 | mtbii 315 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → ¬ 𝑧 ∈ 𝑥) |
7 | 6 | sps 2202 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ 𝑧 ∈ 𝑥) |
8 | 7 | pm2.21d 118 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)) |
9 | 3, 8 | alrimi 2229 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)) |
10 | 9 | anim2i 594 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑥 = 𝑧) → (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) |
11 | 10 | expcom 450 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))) |
12 | 2, 11 | eximd 2232 | . 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 383 ∀wal 1630 ∃wex 1853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 ax-reg 8664 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-v 3342 df-dif 3718 df-un 3720 df-nul 4059 df-sn 4322 df-pr 4324 |
This theorem is referenced by: axregndlem2 9637 axregnd 9638 |
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