Theorem ordtri2or2exmidlem 4671

Description: A set which is 2_o if 𝜑 or ∅ if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)

Assertion

Ref	Expression
ordtri2or2exmidlem	⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On

Distinct variable group:   𝜑,𝑥

Proof of Theorem ordtri2or2exmidlem

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

Step	Hyp	Ref	Expression
1		simpll 531	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ 𝑧)
2		noel 3525	. . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅
3		eleq2 2302	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
4	2, 3	mtbiri 686	. . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑦 ∈ 𝑧)
5	4	adantl 277	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → ¬ 𝑦 ∈ 𝑧)
6	1, 5	pm2.21dd 629	. . . . . 6 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
7		eleq2 2302	. . . . . . . . . . 11 ⊢ (𝑧 = {∅} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {∅}))
8	7	biimpac 298	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅})
9		velsn 3725	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
10	8, 9	sylib 122	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 = ∅)
11		orc 724	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 = ∅ ∨ 𝑦 = {∅}))
12		vex 2824	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
13	12	elpr 3729	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅, {∅}} ↔ (𝑦 = ∅ ∨ 𝑦 = {∅}))
14	11, 13	sylibr 134	. . . . . . . . 9 ⊢ (𝑦 = ∅ → 𝑦 ∈ {∅, {∅}})
15	10, 14	syl 14	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
16	15	adantlr 481	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
17		biidd 172	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜑))
18	17	elrab 2982	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑧 ∈ {∅, {∅}} ∧ 𝜑))
19	18	simprbi 275	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
20	19	ad2antlr 493	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝜑)
21		biidd 172	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
22	21	elrab 2982	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑦 ∈ {∅, {∅}} ∧ 𝜑))
23	16, 20, 22	sylanbrc 421	. . . . . 6 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
24		elrabi 2979	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝑧 ∈ {∅, {∅}})
25		vex 2824	. . . . . . . . 9 ⊢ 𝑧 ∈ V
26	25	elpr 3729	. . . . . . . 8 ⊢ (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
27	24, 26	sylib 122	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
28	27	adantl 277	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
29	6, 23, 28	mpjaodan 810	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
30	29	gen2 1503	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
31		dftr2 4229	. . . 4 ⊢ (Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}))
32	30, 31	mpbir 146	. . 3 ⊢ Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
33		ssrab2 3333	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}}
34		2ordpr 4669	. . 3 ⊢ Ord {∅, {∅}}
35		trssord 4523	. . 3 ⊢ ((Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∧ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
36	32, 33, 34, 35	mp3an 1378	. 2 ⊢ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
37		pp0ex 4324	. . . 4 ⊢ {∅, {∅}} ∈ V
38	37	rabex 4278	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ V
39	38	elon 4517	. 2 ⊢ ({𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
40	36, 39	mpbir 146	1 ⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  ∀wal 1400   = wceq 1402   ∈ wcel 2209  {crab 2532   ⊆ wss 3220  ∅c0 3520  {csn 3708  {cpr 3709  Tr wtr 4227  Ord word 4505  Oncon0 4506

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-uni 3934  df-tr 4228  df-iord 4509  df-on 4511  df-suc 4514

This theorem is referenced by:  ordtri2or2exmid  4716  ontri2orexmidim  4717