Theorem uniss2 3611

Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)

Assertion

Ref	Expression
uniss2	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem uniss2

Step	Hyp	Ref	Expression
1		ssuni 3602	. . . . 5 ⊢ ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ⊆ ∪ 𝐵)
2	1	expcom 109	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵))
3	2	rexlimiv 2427	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → 𝑥 ⊆ ∪ 𝐵)
4	3	ralimi 2384	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵)
5		unissb 3610	. 2 ⊢ (∪ 𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ 𝐵)
6	4, 5	sylibr 137	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307   ⊆ wss 2917  ∪ cuni 3580

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581

This theorem is referenced by:  unidif  3612