Theorem exmid1dc 4233

Description: A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4226 or ordtriexmid 4557. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)

Hypothesis

Ref	Expression
exmid1dc.x	⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})

Assertion

Ref	Expression
exmid1dc	⊢ (𝜑 → EXMID)

Distinct variable group:   𝜑,𝑥

Proof of Theorem exmid1dc

Step	Hyp	Ref	Expression
1		exmid1dc.x	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})
2		exmiddc 837	. . . . . . 7 ⊢ (DECID 𝑥 = {∅} → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
3	1, 2	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ ¬ 𝑥 = {∅}))
4		df-ne 2368	. . . . . . . . 9 ⊢ (𝑥 ≠ {∅} ↔ ¬ 𝑥 = {∅})
5		pwntru 4232	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ {∅} ∧ 𝑥 ≠ {∅}) → 𝑥 = ∅)
6	5	ex 115	. . . . . . . . 9 ⊢ (𝑥 ⊆ {∅} → (𝑥 ≠ {∅} → 𝑥 = ∅))
7	4, 6	biimtrrid 153	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = {∅} → 𝑥 = ∅))
8	7	orim2d 789	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
9	8	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → ((𝑥 = {∅} ∨ ¬ 𝑥 = {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅)))
10	3, 9	mpd 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = {∅} ∨ 𝑥 = ∅))
11	10	orcomd 730	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
12	11	ex 115	. . 3 ⊢ (𝜑 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
13	12	alrimiv 1888	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14		exmid01 4231	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
15	13, 14	sylibr 134	1 ⊢ (𝜑 → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835  ∀wal 1362   = wceq 1364   ≠ wne 2367   ⊆ wss 3157  ∅c0 3450  {csn 3622  EXMIDwem 4227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-exmid 4228

This theorem is referenced by:  pw1fin  6971  exmidonfin  7261  exmidaclem  7275  exmidontri  7306  exmidontri2or  7310