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Mirrors > Home > MPE Home > Th. List > dtruALT | Structured version Visualization version GIF version |
Description: Alternate proof of dtru 5167
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 3549 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 5155 | . . . 4 ⊢ (𝑦 = ∅ → ¬ 𝑦 = {∅}) | |
2 | p0ex 5180 | . . . . 5 ⊢ {∅} ∈ V | |
3 | eqeq2 2806 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑦 = 𝑥 ↔ 𝑦 = {∅})) | |
4 | 3 | notbid 319 | . . . . 5 ⊢ (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅})) |
5 | 2, 4 | spcev 3549 | . . . 4 ⊢ (¬ 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
7 | 0ex 5107 | . . . 4 ⊢ ∅ ∈ V | |
8 | eqeq2 2806 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑦 = 𝑥 ↔ 𝑦 = ∅)) | |
9 | 8 | notbid 319 | . . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅)) |
10 | 7, 9 | spcev 3549 | . . 3 ⊢ (¬ 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
11 | 6, 10 | pm2.61i 183 | . 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥 |
12 | exnal 1808 | . . 3 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥) | |
13 | eqcom 2802 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
14 | 13 | albii 1801 | . . 3 ⊢ (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
15 | 12, 14 | xchbinx 335 | . 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
16 | 11, 15 | mpbi 231 | 1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1520 = wceq 1522 ∃wex 1761 ∅c0 4215 {csn 4476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-v 3439 df-dif 3866 df-in 3870 df-ss 3878 df-nul 4216 df-pw 4459 df-sn 4477 |
This theorem is referenced by: (None) |
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