Theorem reg2exmid 4451

Description: If any inhabited set has a minimal element (when expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)

Hypothesis

Ref	Expression
reg2exmid.1	⊢ ∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)

Assertion

Ref	Expression
reg2exmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable groups:   𝜑,𝑤,𝑧   𝜑,𝑥,𝑧,𝑦

Proof of Theorem reg2exmid

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2139	. . . 4 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
2	1	regexmidlemm 4447	. . 3 ⊢ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
3		reg2exmid.1	. . . 4 ⊢ ∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)
4		pp0ex 4113	. . . . . 6 ⊢ {∅, {∅}} ∈ V
5	4	rabex 4072	. . . . 5 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∈ V
6		eleq2 2203	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
7	6	exbidv 1797	. . . . . 6 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
8		raleq 2626	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦))
9	8	rexeqbi1dv 2635	. . . . . 6 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦 ↔ ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦))
10	7, 9	imbi12d 233	. . . . 5 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ((∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦) ↔ (∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦)))
11	5, 10	spcv 2779	. . . 4 ⊢ (∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦) → (∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦))
12	3, 11	ax-mp 5	. . 3 ⊢ (∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦)
13	2, 12	ax-mp 5	. 2 ⊢ ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦
14	1	reg2exmidlema 4449	. 2 ⊢ (∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦 → (𝜑 ∨ ¬ 𝜑))
15	13, 14	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420   ⊆ wss 3071  ∅c0 3363  {csn 3527  {cpr 3528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534

This theorem is referenced by: (None)