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Theorem ssequn1 3110
Description: A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssequn1 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)

Proof of Theorem ssequn1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 bicom 128 . . . 4 ((𝑥𝐵 ↔ (𝑥𝐴𝑥𝐵)) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐵))
2 pm4.72 736 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵 ↔ (𝑥𝐴𝑥𝐵)))
3 elun 3081 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
43bibi1i 217 . . . 4 ((𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐵))
51, 2, 43bitr4i 201 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
65albii 1359 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
7 dfss2 2931 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
8 dfcleq 2034 . 2 ((𝐴𝐵) = 𝐵 ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐵))
96, 7, 83bitr4i 201 1 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wo 629  wal 1241   = wceq 1243  wcel 1393  cun 2912  wss 2914
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919  df-in 2921  df-ss 2928
This theorem is referenced by:  ssequn2  3113  uniop  3988  pwssunim  4017  unisuc  4121  unisucg  4122  rdgisucinc  5933  oasuc  6005  omsuc  6012
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