Theorem ssfilem 7033

Description: Lemma for ssfiexmid 7034. (Contributed by Jim Kingdon, 3-Feb-2022.)

Hypothesis

Ref	Expression
ssfilem.1	⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin

Assertion

Ref	Expression
ssfilem	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑧

Proof of Theorem ssfilem

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssfilem.1	. . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin
2		isfi 6910	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin ↔ ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛)
3	1, 2	mpbi 145	. 2 ⊢ ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛
4		0elnn 4710	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
5		breq2 4086	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ≈ ∅))
6		en0 6945	. . . . . . . . . 10 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ≈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
7	5, 6	bitrdi 196	. . . . . . . . 9 ⊢ (𝑛 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
8	7	biimpac 298	. . . . . . . 8 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ 𝑛 = ∅) → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
9		rabeq0 3521	. . . . . . . . 9 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ↔ ∀𝑧 ∈ {∅} ¬ 𝜑)
10		0ex 4210	. . . . . . . . . . 11 ⊢ ∅ ∈ V
11	10	snm 3786	. . . . . . . . . 10 ⊢ ∃𝑤 𝑤 ∈ {∅}
12		r19.3rmv 3582	. . . . . . . . . 10 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ 𝜑 ↔ ∀𝑧 ∈ {∅} ¬ 𝜑))
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ (¬ 𝜑 ↔ ∀𝑧 ∈ {∅} ¬ 𝜑)
14	9, 13	bitr4i 187	. . . . . . . 8 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ↔ ¬ 𝜑)
15	8, 14	sylib 122	. . . . . . 7 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ 𝑛 = ∅) → ¬ 𝜑)
16	15	olcd 739	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ 𝑛 = ∅) → (𝜑 ∨ ¬ 𝜑))
17		ensym 6931	. . . . . . . 8 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 → 𝑛 ≈ {𝑧 ∈ {∅} ∣ 𝜑})
18		elex2 2816	. . . . . . . 8 ⊢ (∅ ∈ 𝑛 → ∃𝑥 𝑥 ∈ 𝑛)
19		enm 6975	. . . . . . . 8 ⊢ ((𝑛 ≈ {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∃𝑥 𝑥 ∈ 𝑛) → ∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑})
20	17, 18, 19	syl2an 289	. . . . . . 7 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑})
21		biidd 172	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜑))
22	21	elrab 2959	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (𝑦 ∈ {∅} ∧ 𝜑))
23	22	simprbi 275	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
24	23	orcd 738	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} → (𝜑 ∨ ¬ 𝜑))
25	24	exlimiv 1644	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} → (𝜑 ∨ ¬ 𝜑))
26	20, 25	syl 14	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (𝜑 ∨ ¬ 𝜑))
27	16, 26	jaodan 802	. . . . 5 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (𝜑 ∨ ¬ 𝜑))
28	4, 27	sylan2 286	. . . 4 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ 𝑛 ∈ ω) → (𝜑 ∨ ¬ 𝜑))
29	28	ancoms 268	. . 3 ⊢ ((𝑛 ∈ ω ∧ {𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛) → (𝜑 ∨ ¬ 𝜑))
30	29	rexlimiva 2643	. 2 ⊢ (∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 → (𝜑 ∨ ¬ 𝜑))
31	3, 30	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {crab 2512  ∅c0 3491  {csn 3666   class class class wbr 4082  ωcom 4681   ≈ cen 6883  Fincfn 6885

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4383  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886  df-fin 6888

This theorem is referenced by:  ssfiexmid  7034  domfiexmid  7036