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Theorem sess2 4228
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 3129 . . 3 (𝐴𝐵 → (∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐵𝑦𝑅𝑥} ∈ V))
2 rabss2 3148 . . . . 5 (𝐴𝐵 → {𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥})
3 ssexg 4035 . . . . . 6 (({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥} ∧ {𝑦𝐵𝑦𝑅𝑥} ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
43ex 114 . . . . 5 ({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥} → ({𝑦𝐵𝑦𝑅𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
52, 4syl 14 . . . 4 (𝐴𝐵 → ({𝑦𝐵𝑦𝑅𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
65ralimdv 2475 . . 3 (𝐴𝐵 → (∀𝑥𝐴 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
71, 6syld 45 . 2 (𝐴𝐵 → (∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
8 df-se 4223 . 2 (𝑅 Se 𝐵 ↔ ∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V)
9 df-se 4223 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 93imtr4g 204 1 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  wral 2391  {crab 2395  Vcvv 2658  wss 3039   class class class wbr 3897   Se wse 4219
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  ax-sep 4014
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-rab 2400  df-v 2660  df-in 3045  df-ss 3052  df-se 4223
This theorem is referenced by:  seeq2  4230
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