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Mirrors > Home > ILE Home > Th. List > dtruex | GIF version |
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3933 can also be summarized as "at least two sets exist", the difference is that dtruarb 3933 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific y, we can construct a set x which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
dtruex | ⊢ ∃x ¬ x = y |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 | . . . . 5 ⊢ y ∈ V | |
2 | snexgOLD 3926 | . . . . 5 ⊢ (y ∈ V → {y} ∈ V) | |
3 | 1, 2 | ax-mp 7 | . . . 4 ⊢ {y} ∈ V |
4 | isset 2555 | . . . 4 ⊢ ({y} ∈ V ↔ ∃x x = {y}) | |
5 | 3, 4 | mpbi 133 | . . 3 ⊢ ∃x x = {y} |
6 | elirr 4224 | . . . . . . . 8 ⊢ ¬ y ∈ y | |
7 | ssnid 3395 | . . . . . . . . 9 ⊢ y ∈ {y} | |
8 | eleq2 2098 | . . . . . . . . 9 ⊢ (y = {y} → (y ∈ y ↔ y ∈ {y})) | |
9 | 7, 8 | mpbiri 157 | . . . . . . . 8 ⊢ (y = {y} → y ∈ y) |
10 | 6, 9 | mto 587 | . . . . . . 7 ⊢ ¬ y = {y} |
11 | eqtr 2054 | . . . . . . 7 ⊢ ((y = x ∧ x = {y}) → y = {y}) | |
12 | 10, 11 | mto 587 | . . . . . 6 ⊢ ¬ (y = x ∧ x = {y}) |
13 | ancom 253 | . . . . . 6 ⊢ ((y = x ∧ x = {y}) ↔ (x = {y} ∧ y = x)) | |
14 | 12, 13 | mtbi 594 | . . . . 5 ⊢ ¬ (x = {y} ∧ y = x) |
15 | 14 | imnani 624 | . . . 4 ⊢ (x = {y} → ¬ y = x) |
16 | 15 | eximi 1488 | . . 3 ⊢ (∃x x = {y} → ∃x ¬ y = x) |
17 | 5, 16 | ax-mp 7 | . 2 ⊢ ∃x ¬ y = x |
18 | eqcom 2039 | . . . 4 ⊢ (y = x ↔ x = y) | |
19 | 18 | notbii 593 | . . 3 ⊢ (¬ y = x ↔ ¬ x = y) |
20 | 19 | exbii 1493 | . 2 ⊢ (∃x ¬ y = x ↔ ∃x ¬ x = y) |
21 | 17, 20 | mpbi 133 | 1 ⊢ ∃x ¬ x = y |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 {csn 3367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-v 2553 df-dif 2914 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 |
This theorem is referenced by: dtru 4238 eunex 4239 brprcneu 5114 |
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