Theorem dfon2lem1

Description: Lemma for dfon2 . (Contributed by Scott Fenton, 28-Feb-2011) (Proof shortened by SN, 5-Oct-2026)

	Ref	Expression
Assertion	dfon2lem1	⊢ Tr ∪ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) }

Proof

Step	Hyp	Ref	Expression
1		truni	⊢ ( ∀ 𝑦 ∈ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } Tr 𝑦 → Tr ∪ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } )
2		simp2	⊢ ( ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) → Tr 𝑥 )
3	2	ss2abi	⊢ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } ⊆ { 𝑥 ∣ Tr 𝑥 }
4	3	sseli	⊢ ( 𝑦 ∈ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } → 𝑦 ∈ { 𝑥 ∣ Tr 𝑥 } )
5		vex	⊢ 𝑦 ∈ V
6		treq	⊢ ( 𝑥 = 𝑦 → ( Tr 𝑥 ↔ Tr 𝑦 ) )
7	5 6	elab	⊢ ( 𝑦 ∈ { 𝑥 ∣ Tr 𝑥 } ↔ Tr 𝑦 )
8	4 7	sylib	⊢ ( 𝑦 ∈ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } → Tr 𝑦 )
9	1 8	mprg	⊢ Tr ∪ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) }