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Mirrors > Home > MPE Home > Th. List > dtruALT | Structured version Visualization version GIF version |
Description: Alternate proof of dtru 4887
which requires more axioms but is shorter and
may be easier to understand.
Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, theorem spcev 3331 requires that 𝑥 must not occur in the subexpression ¬ 𝑦 = {∅} in step 4 nor in the subexpression ¬ 𝑦 = ∅ in step 9. The proof verifier will require that 𝑥 and 𝑦 be in a distinct variable group to ensure this. You can check this by deleting the $d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dtruALT | ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0inp0 4867 | . . . 4 ⊢ (𝑦 = ∅ → ¬ 𝑦 = {∅}) | |
2 | p0ex 4883 | . . . . 5 ⊢ {∅} ∈ V | |
3 | eqeq2 2662 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑦 = 𝑥 ↔ 𝑦 = {∅})) | |
4 | 3 | notbid 307 | . . . . 5 ⊢ (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅})) |
5 | 2, 4 | spcev 3331 | . . . 4 ⊢ (¬ 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
7 | 0ex 4823 | . . . 4 ⊢ ∅ ∈ V | |
8 | eqeq2 2662 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑦 = 𝑥 ↔ 𝑦 = ∅)) | |
9 | 8 | notbid 307 | . . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅)) |
10 | 7, 9 | spcev 3331 | . . 3 ⊢ (¬ 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
11 | 6, 10 | pm2.61i 176 | . 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥 |
12 | exnal 1794 | . . 3 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥) | |
13 | eqcom 2658 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
14 | 13 | albii 1787 | . . 3 ⊢ (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
15 | 12, 14 | xchbinx 323 | . 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
16 | 11, 15 | mpbi 220 | 1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1521 = wceq 1523 ∃wex 1744 ∅c0 3948 {csn 4210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-v 3233 df-dif 3610 df-in 3614 df-ss 3621 df-nul 3949 df-pw 4193 df-sn 4211 |
This theorem is referenced by: (None) |
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