Theorem trintssmOLD 3892

Description: Obsolete version of trintssm 3891 as of 30-Oct-2021. (Contributed by Jim Kingdon, 22-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
trintssmOLD	⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → ∩ 𝐴 ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem trintssmOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . 4 ⊢ 𝑦 ∈ V
2	1	elint2 3643	. . 3 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
3		r19.2m 3329	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
4	3	ex 113	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
5		trel 3882	. . . . . 6 ⊢ (Tr 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
6	5	expcomd 1370	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
7	6	rexlimdv 2476	. . . 4 ⊢ (Tr 𝐴 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
8	4, 7	sylan9 401	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
9	2, 8	syl5bi 150	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → (𝑦 ∈ ∩ 𝐴 → 𝑦 ∈ 𝐴))
10	9	ssrdv 3005	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ Tr 𝐴) → ∩ 𝐴 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 102  ∃wex 1421   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349   ⊆ wss 2973  ∩ cint 3636  Tr wtr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-int 3637  df-tr 3876

This theorem is referenced by: (None)