Theorem bj-elssuniab 9199

Description: Version of elssuni 3599 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

Hypothesis

Ref	Expression
bj-elssuniab.nf	⊢ ℲxA

Assertion

Ref	Expression
bj-elssuniab	⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ}))

Proof of Theorem bj-elssuniab

Step	Hyp	Ref	Expression
1		sbc8g 2765	. 2 ⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ}))
2		elssuni 3599	. 2 ⊢ (A ∈ {x ∣ φ} → A ⊆ ∪ {x ∣ φ})
3	1, 2	syl6bi 152	1 ⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ}))

Syntax hints:   → wi 4   ∈ wcel 1390  {cab 2023  Ⅎwnfc 2162  [wsbc 2758   ⊆ wss 2911  ∪ cuni 3571

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-in 2918  df-ss 2925  df-uni 3572

This theorem is referenced by: (None)