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Theorem unissb 3734
Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
unissb ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unissb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eluni 3707 . . . . . 6 (𝑦 𝐴 ↔ ∃𝑥(𝑦𝑥𝑥𝐴))
21imbi1i 237 . . . . 5 ((𝑦 𝐴𝑦𝐵) ↔ (∃𝑥(𝑦𝑥𝑥𝐴) → 𝑦𝐵))
3 19.23v 1837 . . . . 5 (∀𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐵) ↔ (∃𝑥(𝑦𝑥𝑥𝐴) → 𝑦𝐵))
42, 3bitr4i 186 . . . 4 ((𝑦 𝐴𝑦𝐵) ↔ ∀𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐵))
54albii 1429 . . 3 (∀𝑦(𝑦 𝐴𝑦𝐵) ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐵))
6 alcom 1437 . . . 4 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐵) ↔ ∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐵))
7 19.21v 1827 . . . . . 6 (∀𝑦(𝑥𝐴 → (𝑦𝑥𝑦𝐵)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))
8 impexp 261 . . . . . . . 8 (((𝑦𝑥𝑥𝐴) → 𝑦𝐵) ↔ (𝑦𝑥 → (𝑥𝐴𝑦𝐵)))
9 bi2.04 247 . . . . . . . 8 ((𝑦𝑥 → (𝑥𝐴𝑦𝐵)) ↔ (𝑥𝐴 → (𝑦𝑥𝑦𝐵)))
108, 9bitri 183 . . . . . . 7 (((𝑦𝑥𝑥𝐴) → 𝑦𝐵) ↔ (𝑥𝐴 → (𝑦𝑥𝑦𝐵)))
1110albii 1429 . . . . . 6 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐵) ↔ ∀𝑦(𝑥𝐴 → (𝑦𝑥𝑦𝐵)))
12 dfss2 3054 . . . . . . 7 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
1312imbi2i 225 . . . . . 6 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))
147, 11, 133bitr4i 211 . . . . 5 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐵) ↔ (𝑥𝐴𝑥𝐵))
1514albii 1429 . . . 4 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
166, 15bitri 183 . . 3 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
175, 16bitri 183 . 2 (∀𝑦(𝑦 𝐴𝑦𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
18 dfss2 3054 . 2 ( 𝐴𝐵 ↔ ∀𝑦(𝑦 𝐴𝑦𝐵))
19 df-ral 2396 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2017, 18, 193bitr4i 211 1 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312  wex 1451  wcel 1463  wral 2391  wss 3039   cuni 3704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-in 3045  df-ss 3052  df-uni 3705
This theorem is referenced by:  uniss2  3735  ssunieq  3737  sspwuni  3865  pwssb  3866  bm2.5ii  4380  sbthlem1  6811  neipsm  12218  neiuni  12225
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