Theorem rabxmdc 3249

Description: Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)

Assertion

Ref	Expression
rabxmdc	⊢ (∀𝑥DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabxmdc

Step	Hyp	Ref	Expression
1		exmiddc 744	. . . . . 6 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2	1	a1d 22	. . . . 5 ⊢ (DECID 𝜑 → (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
3	2	alimi 1344	. . . 4 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
4		df-ral 2311	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑)))
5	3, 4	sylibr 137	. . 3 ⊢ (∀𝑥DECID 𝜑 → ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑))
6		rabid2 2486	. . 3 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑))
7	5, 6	sylibr 137	. 2 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
8		unrab 3208	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
9	7, 8	syl6eqr 2090	1 ⊢ (∀𝑥DECID 𝜑 → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629  DECID wdc 742  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310   ∪ cun 2915

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-dc 743  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-rab 2315  df-v 2559  df-un 2922

This theorem is referenced by: (None)