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Mirrors > Home > MPE Home > Th. List > dtruALT | Structured version Visualization version GIF version |
Description: Alternate proof of dtru 5429
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 3590 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 5350 | . . . 4 ⊢ (𝑦 = ∅ → ¬ 𝑦 = {∅}) | |
2 | p0ex 5375 | . . . . 5 ⊢ {∅} ∈ V | |
3 | eqeq2 2738 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑦 = 𝑥 ↔ 𝑦 = {∅})) | |
4 | 3 | notbid 318 | . . . . 5 ⊢ (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅})) |
5 | 2, 4 | spcev 3590 | . . . 4 ⊢ (¬ 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
7 | 0ex 5300 | . . . 4 ⊢ ∅ ∈ V | |
8 | eqeq2 2738 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑦 = 𝑥 ↔ 𝑦 = ∅)) | |
9 | 8 | notbid 318 | . . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅)) |
10 | 7, 9 | spcev 3590 | . . 3 ⊢ (¬ 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥) |
11 | 6, 10 | pm2.61i 182 | . 2 ⊢ ∃𝑥 ¬ 𝑦 = 𝑥 |
12 | exnal 1821 | . . 3 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥) | |
13 | eqcom 2733 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
14 | 13 | albii 1813 | . . 3 ⊢ (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) |
15 | 12, 14 | xchbinx 334 | . 2 ⊢ (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦) |
16 | 11, 15 | mpbi 229 | 1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1531 = wceq 1533 ∃wex 1773 ∅c0 4317 {csn 4623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-dif 3946 df-in 3950 df-ss 3960 df-nul 4318 df-pw 4599 df-sn 4624 |
This theorem is referenced by: (None) |
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