Theorem onsucelsucexmidlem 4565

Description: Lemma for onsucelsucexmid 4566. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5913), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4555. (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 3454	. . . . . . . . . 10 ⊢ ¬ 𝑦 ∈ ∅
3		eleq2 2260	. . . . . . . . . 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 2260	. . . . . . . . . . 11 ⊢ (𝑧 = {∅} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {∅}))
9	8	biimpac 298	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅})
10		velsn 3639	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
11	9, 10	sylib 122	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 = ∅)
12		onsucelsucexmidlem1 4564	. . . . . . . . 9 ⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
13	11, 12	eqeltrdi 2287	. . . . . . . 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 2917	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → 𝑧 ∈ {∅, {∅}})
17		vex 2766	. . . . . . . . 9 ⊢ 𝑧 ∈ V
18	17	elpr 3643	. . . . . . . 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 1464	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
23		dftr2 4133	. . . 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 3268	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}}
26		2ordpr 4560	. . 3 ⊢ Ord {∅, {∅}}
27		trssord 4415	. . 3 ⊢ ((Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
28	24, 25, 26, 27	mp3an 1348	. 2 ⊢ Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
29		pp0ex 4222	. . . 4 ⊢ {∅, {∅}} ∈ V
30	29	rabex 4177	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ V
31	30	elon 4409	. 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 1362   = wceq 1364   ∈ wcel 2167  {crab 2479   ⊆ wss 3157  ∅c0 3450  {csn 3622  {cpr 3623  Tr wtr 4131  Ord word 4397  Oncon0 4398

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-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406

This theorem is referenced by:  onsucelsucexmid  4566  acexmidlemcase  5917  acexmidlemv  5920