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

Theorem sess2 5517
 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 4031 . . 3 (𝐴𝐵 → (∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐵𝑦𝑅𝑥} ∈ V))
2 rabss2 4052 . . . . 5 (𝐴𝐵 → {𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥})
3 ssexg 5218 . . . . . 6 (({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥} ∧ {𝑦𝐵𝑦𝑅𝑥} ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
43ex 415 . . . . 5 ({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥} → ({𝑦𝐵𝑦𝑅𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
52, 4syl 17 . . . 4 (𝐴𝐵 → ({𝑦𝐵𝑦𝑅𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
65ralimdv 3176 . . 3 (𝐴𝐵 → (∀𝑥𝐴 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
71, 6syld 47 . 2 (𝐴𝐵 → (∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
8 df-se 5508 . 2 (𝑅 Se 𝐵 ↔ ∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V)
9 df-se 5508 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 93imtr4g 298 1 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2107  ∀wral 3136  {crab 3140  Vcvv 3493   ⊆ wss 3934   class class class wbr 5057   Se wse 5505 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rab 3145  df-v 3495  df-in 3941  df-ss 3950  df-se 5508 This theorem is referenced by:  seeq2  5521  wereu2  5545  wfrlem5  7951  frpomin  33066  frmin  33072  fprlem1  33125  frrlem15  33130
 Copyright terms: Public domain W3C validator