Theorem onsucelsucexmidlem 4575

Description: Lemma for onsucelsucexmid 4576. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5925), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4565. (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 3463	. . . . . . . . . 10 ⊢ ¬ 𝑦 ∈ ∅
3		eleq2 2268	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
4	2, 3	mtbiri 676	. . . . . . . . 9 ⊢ (𝑧 = ∅ → ¬ 𝑦 ∈ 𝑧)
5	4	adantl 277	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → ¬ 𝑦 ∈ 𝑧)
6	1, 5	pm2.21dd 621	. . . . . . 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 2268	. . . . . . . . . . 11 ⊢ (𝑧 = {∅} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {∅}))
9	8	biimpac 298	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅})
10		velsn 3649	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
11	9, 10	sylib 122	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 = ∅)
12		onsucelsucexmidlem1 4574	. . . . . . . . 9 ⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
13	11, 12	eqeltrdi 2295	. . . . . . . 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 2925	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → 𝑧 ∈ {∅, {∅}})
17		vex 2774	. . . . . . . . 9 ⊢ 𝑧 ∈ V
18	17	elpr 3653	. . . . . . . 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 719	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
22	21	gen2 1472	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
23		dftr2 4143	. . . 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 3277	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}}
26		2ordpr 4570	. . 3 ⊢ Ord {∅, {∅}}
27		trssord 4425	. . 3 ⊢ ((Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
28	24, 25, 26, 27	mp3an 1349	. 2 ⊢ Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
29		pp0ex 4232	. . . 4 ⊢ {∅, {∅}} ∈ V
30	29	rabex 4187	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ V
31	30	elon 4419	. 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 709  ∀wal 1370   = wceq 1372   ∈ wcel 2175  {crab 2487   ⊆ wss 3165  ∅c0 3459  {csn 3632  {cpr 3633  Tr wtr 4141  Ord word 4407  Oncon0 4408

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-tr 4142  df-iord 4411  df-on 4413  df-suc 4416

This theorem is referenced by:  onsucelsucexmid  4576  acexmidlemcase  5929  acexmidlemv  5932