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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-zfpair2 | GIF version |
Description: Proof of zfpair2 3936 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-zfpair2 | ⊢ {x, y} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bdeq 9275 | . . . . 5 ⊢ BOUNDED w = x | |
2 | ax-bdeq 9275 | . . . . 5 ⊢ BOUNDED w = y | |
3 | 1, 2 | ax-bdor 9271 | . . . 4 ⊢ BOUNDED (w = x ∨ w = y) |
4 | ax-pr 3935 | . . . 4 ⊢ ∃z∀w((w = x ∨ w = y) → w ∈ z) | |
5 | 3, 4 | bdbm1.3ii 9346 | . . 3 ⊢ ∃z∀w(w ∈ z ↔ (w = x ∨ w = y)) |
6 | dfcleq 2031 | . . . . 5 ⊢ (z = {x, y} ↔ ∀w(w ∈ z ↔ w ∈ {x, y})) | |
7 | vex 2554 | . . . . . . . 8 ⊢ w ∈ V | |
8 | 7 | elpr 3385 | . . . . . . 7 ⊢ (w ∈ {x, y} ↔ (w = x ∨ w = y)) |
9 | 8 | bibi2i 216 | . . . . . 6 ⊢ ((w ∈ z ↔ w ∈ {x, y}) ↔ (w ∈ z ↔ (w = x ∨ w = y))) |
10 | 9 | albii 1356 | . . . . 5 ⊢ (∀w(w ∈ z ↔ w ∈ {x, y}) ↔ ∀w(w ∈ z ↔ (w = x ∨ w = y))) |
11 | 6, 10 | bitri 173 | . . . 4 ⊢ (z = {x, y} ↔ ∀w(w ∈ z ↔ (w = x ∨ w = y))) |
12 | 11 | exbii 1493 | . . 3 ⊢ (∃z z = {x, y} ↔ ∃z∀w(w ∈ z ↔ (w = x ∨ w = y))) |
13 | 5, 12 | mpbir 134 | . 2 ⊢ ∃z z = {x, y} |
14 | 13 | issetri 2558 | 1 ⊢ {x, y} ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∨ wo 628 ∀wal 1240 = wceq 1242 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 {cpr 3368 |
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-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-pr 3935 ax-bdor 9271 ax-bdeq 9275 ax-bdsep 9339 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 |
This theorem is referenced by: bj-prexg 9366 |
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