Theorem onsucelsucexmidlem 4633

Description: Lemma for onsucelsucexmid 4634. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 6019), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4623. (Contributed by Jim Kingdon, 2-Aug-2019.)

Assertion

Ref	Expression
onsucelsucexmidlem	⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On

Distinct variable group:   𝜑,𝑥

Proof of Theorem onsucelsucexmidlem

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → 𝑦 ∈ 𝑧)
2		noel 3500	. . . . . . . . . 10 ⊢ ¬ 𝑦 ∈ ∅
3		eleq2 2295	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
4	2, 3	mtbiri 682	. . . . . . . . 9 ⊢ (𝑧 = ∅ → ¬ 𝑦 ∈ 𝑧)
5	4	adantl 277	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → ¬ 𝑦 ∈ 𝑧)
6	1, 5	pm2.21dd 625	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
7	6	ex 115	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = ∅ → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
8		eleq2 2295	. . . . . . . . . . 11 ⊢ (𝑧 = {∅} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {∅}))
9	8	biimpac 298	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅})
10		velsn 3690	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
11	9, 10	sylib 122	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 = ∅)
12		onsucelsucexmidlem1 4632	. . . . . . . . 9 ⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
13	11, 12	eqeltrdi 2322	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
14	13	ex 115	. . . . . . 7 ⊢ (𝑦 ∈ 𝑧 → (𝑧 = {∅} → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
15	14	adantr 276	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = {∅} → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
16		elrabi 2960	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → 𝑧 ∈ {∅, {∅}})
17		vex 2806	. . . . . . . . 9 ⊢ 𝑧 ∈ V
18	17	elpr 3694	. . . . . . . 8 ⊢ (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
19	16, 18	sylib 122	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
20	19	adantl 277	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
21	7, 15, 20	mpjaod 726	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
22	21	gen2 1499	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
23		dftr2 4194	. . . 4 ⊢ (Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
24	22, 23	mpbir 146	. . 3 ⊢ Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
25		ssrab2 3313	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}}
26		2ordpr 4628	. . 3 ⊢ Ord {∅, {∅}}
27		trssord 4483	. . 3 ⊢ ((Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
28	24, 25, 26, 27	mp3an 1374	. 2 ⊢ Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
29		pp0ex 4285	. . . 4 ⊢ {∅, {∅}} ∈ V
30	29	rabex 4239	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ V
31	30	elon 4477	. 2 ⊢ ({𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
32	28, 31	mpbir 146	1 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  ∀wal 1396   = wceq 1398   ∈ wcel 2202  {crab 2515   ⊆ wss 3201  ∅c0 3496  {csn 3673  {cpr 3674  Tr wtr 4192  Ord word 4465  Oncon0 4466

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474

This theorem is referenced by:  onsucelsucexmid  4634  acexmidlemcase  6023  acexmidlemv  6026