MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sess2 Structured version   Visualization version   GIF version

Theorem sess2 5281
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
sess2 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))

Proof of Theorem sess2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssralv 3862 . . 3 (𝐴𝐵 → (∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐵𝑦𝑅𝑥} ∈ V))
2 rabss2 3881 . . . . 5 (𝐴𝐵 → {𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥})
3 ssexg 4999 . . . . . 6 (({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥} ∧ {𝑦𝐵𝑦𝑅𝑥} ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
43ex 402 . . . . 5 ({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥} → ({𝑦𝐵𝑦𝑅𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
52, 4syl 17 . . . 4 (𝐴𝐵 → ({𝑦𝐵𝑦𝑅𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
65ralimdv 3144 . . 3 (𝐴𝐵 → (∀𝑥𝐴 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
71, 6syld 47 . 2 (𝐴𝐵 → (∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
8 df-se 5272 . 2 (𝑅 Se 𝐵 ↔ ∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V)
9 df-se 5272 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 93imtr4g 288 1 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wral 3089  {crab 3093  Vcvv 3385  wss 3769   class class class wbr 4843   Se wse 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-in 3776  df-ss 3783  df-se 5272
This theorem is referenced by:  seeq2  5285  wereu2  5309  wfrlem5  7658  frpomin  32251  frmin  32255  frrlem5  32297
  Copyright terms: Public domain W3C validator