Theorem reg2exmid 4605

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 2209	. . . 4 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
2	1	regexmidlemm 4601	. . 3 ⊢ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
3		reg2exmid.1	. . . 4 ⊢ ∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)
4		pp0ex 4252	. . . . . 6 ⊢ {∅, {∅}} ∈ V
5	4	rabex 4207	. . . . 5 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∈ V
6		eleq2 2273	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
7	6	exbidv 1851	. . . . . 6 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
8		raleq 2708	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦))
9	8	rexeqbi1dv 2721	. . . . . 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 234	. . . . 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 2877	. . . 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 4603	. 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 104   ∨ wo 712  ∀wal 1373   = wceq 1375  ∃wex 1518   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  {crab 2492   ⊆ wss 3177  ∅c0 3471  {csn 3646  {cpr 3647

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653

This theorem is referenced by: (None)