Theorem uni0c 3606

Description: The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Assertion

Ref	Expression
uni0c	⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem uni0c

Step	Hyp	Ref	Expression
1		uni0b 3605	. 2 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2		dfss3 2935	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
3		velsn 3392	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	ralbii 2330	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
5	1, 2, 4	3bitri 195	1 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)

Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ⊆ wss 2917  ∅c0 3224  {csn 3375  ∪ 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-in1 544  ax-in2 545  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-fal 1249  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-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581

This theorem is referenced by: (None)