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Theorem elvvv 4326
Description: Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
elvvv (A ((V × V) × V) ↔ xyz A = ⟨⟨x, y⟩, z⟩)
Distinct variable group:   x,y,z,A

Proof of Theorem elvvv
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 elxp 4285 . 2 (A ((V × V) × V) ↔ wz(A = ⟨w, z (w (V × V) z V)))
2 anass 383 . . . . 5 (((A = ⟨w, z w (V × V)) z V) ↔ (A = ⟨w, z (w (V × V) z V)))
3 19.42vv 1766 . . . . . 6 (xy(A = ⟨w, z w = ⟨x, y⟩) ↔ (A = ⟨w, z xy w = ⟨x, y⟩))
4 ancom 253 . . . . . . 7 ((w = ⟨x, y A = ⟨w, z⟩) ↔ (A = ⟨w, z w = ⟨x, y⟩))
542exbii 1475 . . . . . 6 (xy(w = ⟨x, y A = ⟨w, z⟩) ↔ xy(A = ⟨w, z w = ⟨x, y⟩))
6 vex 2534 . . . . . . . 8 z V
76biantru 286 . . . . . . 7 ((A = ⟨w, z w (V × V)) ↔ ((A = ⟨w, z w (V × V)) z V))
8 elvv 4325 . . . . . . . 8 (w (V × V) ↔ xy w = ⟨x, y⟩)
98anbi2i 433 . . . . . . 7 ((A = ⟨w, z w (V × V)) ↔ (A = ⟨w, z xy w = ⟨x, y⟩))
107, 9bitr3i 175 . . . . . 6 (((A = ⟨w, z w (V × V)) z V) ↔ (A = ⟨w, z xy w = ⟨x, y⟩))
113, 5, 103bitr4ri 202 . . . . 5 (((A = ⟨w, z w (V × V)) z V) ↔ xy(w = ⟨x, y A = ⟨w, z⟩))
122, 11bitr3i 175 . . . 4 ((A = ⟨w, z (w (V × V) z V)) ↔ xy(w = ⟨x, y A = ⟨w, z⟩))
13122exbii 1475 . . 3 (wz(A = ⟨w, z (w (V × V) z V)) ↔ wzxy(w = ⟨x, y A = ⟨w, z⟩))
14 exrot4 1559 . . . 4 (xywz(w = ⟨x, y A = ⟨w, z⟩) ↔ wzxy(w = ⟨x, y A = ⟨w, z⟩))
15 excom 1532 . . . . . 6 (wz(w = ⟨x, y A = ⟨w, z⟩) ↔ zw(w = ⟨x, y A = ⟨w, z⟩))
16 vex 2534 . . . . . . . . 9 x V
17 vex 2534 . . . . . . . . 9 y V
1816, 17opex 3936 . . . . . . . 8 x, y V
19 opeq1 3519 . . . . . . . . 9 (w = ⟨x, y⟩ → ⟨w, z⟩ = ⟨⟨x, y⟩, z⟩)
2019eqeq2d 2029 . . . . . . . 8 (w = ⟨x, y⟩ → (A = ⟨w, z⟩ ↔ A = ⟨⟨x, y⟩, z⟩))
2118, 20ceqsexv 2566 . . . . . . 7 (w(w = ⟨x, y A = ⟨w, z⟩) ↔ A = ⟨⟨x, y⟩, z⟩)
2221exbii 1474 . . . . . 6 (zw(w = ⟨x, y A = ⟨w, z⟩) ↔ z A = ⟨⟨x, y⟩, z⟩)
2315, 22bitri 173 . . . . 5 (wz(w = ⟨x, y A = ⟨w, z⟩) ↔ z A = ⟨⟨x, y⟩, z⟩)
24232exbii 1475 . . . 4 (xywz(w = ⟨x, y A = ⟨w, z⟩) ↔ xyz A = ⟨⟨x, y⟩, z⟩)
2514, 24bitr3i 175 . . 3 (wzxy(w = ⟨x, y A = ⟨w, z⟩) ↔ xyz A = ⟨⟨x, y⟩, z⟩)
2613, 25bitri 173 . 2 (wz(A = ⟨w, z (w (V × V) z V)) ↔ xyz A = ⟨⟨x, y⟩, z⟩)
271, 26bitri 173 1 (A ((V × V) × V) ↔ xyz A = ⟨⟨x, y⟩, z⟩)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1226  wex 1358   wcel 1370  Vcvv 2531  cop 3349   × cxp 4266
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-opab 3789  df-xp 4274
This theorem is referenced by:  ssrelrel  4363  dftpos3  5795
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