Theorem pwle2 12885

Description: An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)

Hypothesis

Ref	Expression
pwle2.t	⊢ 𝑇 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o)

Assertion

Ref	Expression
pwle2	⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → 𝑁 ⊆ 2o)

Distinct variable group:   𝑥,𝑁

Allowed substitution hints:   𝑇(𝑥)   𝐺(𝑥)

Proof of Theorem pwle2

Step	Hyp	Ref	Expression
1		simplr 502	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → 𝐺:𝑇–1-1→𝒫 1o)
2		f1f 5286	. . . . . . . . . . 11 ⊢ (𝐺:𝑇–1-1→𝒫 1o → 𝐺:𝑇⟶𝒫 1o)
3	1, 2	syl 14	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → 𝐺:𝑇⟶𝒫 1o)
4		nnon 4483	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
5	4	ad2antrr 477	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → 𝑁 ∈ On)
6		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → 2o ∈ 𝑁)
7		1lt2o 6293	. . . . . . . . . . . . . 14 ⊢ 1o ∈ 2o
8	6, 7	jctil 308	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (1o ∈ 2o ∧ 2o ∈ 𝑁))
9		ontr1 4271	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ On → ((1o ∈ 2o ∧ 2o ∈ 𝑁) → 1o ∈ 𝑁))
10	5, 8, 9	sylc 62	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → 1o ∈ 𝑁)
11		0lt1o 6291	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1o
12		opelxpi 4531	. . . . . . . . . . . 12 ⊢ ((1o ∈ 𝑁 ∧ ∅ ∈ 1o) → ⟨1o, ∅⟩ ∈ (𝑁 × 1o))
13	10, 11, 12	sylancl 407	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨1o, ∅⟩ ∈ (𝑁 × 1o))
14		pwle2.t	. . . . . . . . . . . 12 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o)
15		iunxpconst 4559	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o) = (𝑁 × 1o)
16	14, 15	eqtri 2135	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑁 × 1o)
17	13, 16	syl6eleqr 2208	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨1o, ∅⟩ ∈ 𝑇)
18	3, 17	ffvelrnd 5510	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨1o, ∅⟩) ∈ 𝒫 1o)
19	18	elpwid 3487	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨1o, ∅⟩) ⊆ 1o)
20		df1o2 6280	. . . . . . . 8 ⊢ 1o = {∅}
21	19, 20	syl6sseq 3111	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨1o, ∅⟩) ⊆ {∅})
22		pwtrufal 12884	. . . . . . 7 ⊢ ((𝐺‘⟨1o, ∅⟩) ⊆ {∅} → ¬ ¬ ((𝐺‘⟨1o, ∅⟩) = ∅ ∨ (𝐺‘⟨1o, ∅⟩) = {∅}))
23	21, 22	syl 14	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ ¬ ((𝐺‘⟨1o, ∅⟩) = ∅ ∨ (𝐺‘⟨1o, ∅⟩) = {∅}))
24		ioran 724	. . . . . 6 ⊢ (¬ ((𝐺‘⟨1o, ∅⟩) = ∅ ∨ (𝐺‘⟨1o, ∅⟩) = {∅}) ↔ (¬ (𝐺‘⟨1o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨1o, ∅⟩) = {∅}))
25	23, 24	sylnib 648	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (¬ (𝐺‘⟨1o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨1o, ∅⟩) = {∅}))
26		1n0 6283	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
27	26	neii 2284	. . . . . . . . . 10 ⊢ ¬ 1o = ∅
28		1oex 6275	. . . . . . . . . . 11 ⊢ 1o ∈ V
29		0ex 4015	. . . . . . . . . . 11 ⊢ ∅ ∈ V
30	28, 29	opth1 4118	. . . . . . . . . 10 ⊢ (⟨1o, ∅⟩ = ⟨∅, ∅⟩ → 1o = ∅)
31	27, 30	mto 634	. . . . . . . . 9 ⊢ ¬ ⟨1o, ∅⟩ = ⟨∅, ∅⟩
32		0lt2o 6292	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ 2o
33	6, 32	jctil 308	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (∅ ∈ 2o ∧ 2o ∈ 𝑁))
34		ontr1 4271	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ On → ((∅ ∈ 2o ∧ 2o ∈ 𝑁) → ∅ ∈ 𝑁))
35	5, 33, 34	sylc 62	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ∅ ∈ 𝑁)
36		opelxpi 4531	. . . . . . . . . . . 12 ⊢ ((∅ ∈ 𝑁 ∧ ∅ ∈ 1o) → ⟨∅, ∅⟩ ∈ (𝑁 × 1o))
37	35, 11, 36	sylancl 407	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨∅, ∅⟩ ∈ (𝑁 × 1o))
38	37, 16	syl6eleqr 2208	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨∅, ∅⟩ ∈ 𝑇)
39		f1veqaeq 5624	. . . . . . . . . 10 ⊢ ((𝐺:𝑇–1-1→𝒫 1o ∧ (⟨1o, ∅⟩ ∈ 𝑇 ∧ ⟨∅, ∅⟩ ∈ 𝑇)) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) → ⟨1o, ∅⟩ = ⟨∅, ∅⟩))
40	1, 17, 38, 39	syl12anc 1197	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) → ⟨1o, ∅⟩ = ⟨∅, ∅⟩))
41	31, 40	mtoi 636	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
42	41	adantr 272	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
43		simpr 109	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → (𝐺‘⟨∅, ∅⟩) = ∅)
44	43	eqeq2d 2126	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) ↔ (𝐺‘⟨1o, ∅⟩) = ∅))
45	42, 44	mtbid 644	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → ¬ (𝐺‘⟨1o, ∅⟩) = ∅)
46		2on0 6277	. . . . . . . . . . . . . 14 ⊢ 2o ≠ ∅
47	46	nesymi 2328	. . . . . . . . . . . . 13 ⊢ ¬ ∅ = 2o
48	29, 29	opth1 4118	. . . . . . . . . . . . 13 ⊢ (⟨∅, ∅⟩ = ⟨2o, ∅⟩ → ∅ = 2o)
49	47, 48	mto 634	. . . . . . . . . . . 12 ⊢ ¬ ⟨∅, ∅⟩ = ⟨2o, ∅⟩
50		opelxpi 4531	. . . . . . . . . . . . . . 15 ⊢ ((2o ∈ 𝑁 ∧ ∅ ∈ 1o) → ⟨2o, ∅⟩ ∈ (𝑁 × 1o))
51	6, 11, 50	sylancl 407	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨2o, ∅⟩ ∈ (𝑁 × 1o))
52	51, 16	syl6eleqr 2208	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨2o, ∅⟩ ∈ 𝑇)
53		f1veqaeq 5624	. . . . . . . . . . . . 13 ⊢ ((𝐺:𝑇–1-1→𝒫 1o ∧ (⟨∅, ∅⟩ ∈ 𝑇 ∧ ⟨2o, ∅⟩ ∈ 𝑇)) → ((𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩) → ⟨∅, ∅⟩ = ⟨2o, ∅⟩))
54	1, 38, 52, 53	syl12anc 1197	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ((𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩) → ⟨∅, ∅⟩ = ⟨2o, ∅⟩))
55	49, 54	mtoi 636	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
56	55	ad2antrr 477	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
57		simplr 502	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → (𝐺‘⟨∅, ∅⟩) = ∅)
58	57	eqeq1d 2123	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ((𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩) ↔ ∅ = (𝐺‘⟨2o, ∅⟩)))
59	56, 58	mtbid 644	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ ∅ = (𝐺‘⟨2o, ∅⟩))
60		eqcom 2117	. . . . . . . . 9 ⊢ (∅ = (𝐺‘⟨2o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = ∅)
61	59, 60	sylnib 648	. . . . . . . 8 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (𝐺‘⟨2o, ∅⟩) = ∅)
62		1onn 6370	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ ω
63		peano1 4468	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ ω
64		peano4 4471	. . . . . . . . . . . . . . . . . 18 ⊢ ((1o ∈ ω ∧ ∅ ∈ ω) → (suc 1o = suc ∅ ↔ 1o = ∅))
65	62, 63, 64	mp2an 420	. . . . . . . . . . . . . . . . 17 ⊢ (suc 1o = suc ∅ ↔ 1o = ∅)
66	26, 65	nemtbir 2371	. . . . . . . . . . . . . . . 16 ⊢ ¬ suc 1o = suc ∅
67		df-2o 6268	. . . . . . . . . . . . . . . . 17 ⊢ 2o = suc 1o
68		df-1o 6267	. . . . . . . . . . . . . . . . 17 ⊢ 1o = suc ∅
69	67, 68	eqeq12i 2128	. . . . . . . . . . . . . . . 16 ⊢ (2o = 1o ↔ suc 1o = suc ∅)
70	66, 69	mtbir 643	. . . . . . . . . . . . . . 15 ⊢ ¬ 2o = 1o
71	70	neir 2285	. . . . . . . . . . . . . 14 ⊢ 2o ≠ 1o
72	71	nesymi 2328	. . . . . . . . . . . . 13 ⊢ ¬ 1o = 2o
73	28, 29	opth1 4118	. . . . . . . . . . . . 13 ⊢ (⟨1o, ∅⟩ = ⟨2o, ∅⟩ → 1o = 2o)
74	72, 73	mto 634	. . . . . . . . . . . 12 ⊢ ¬ ⟨1o, ∅⟩ = ⟨2o, ∅⟩
75		f1veqaeq 5624	. . . . . . . . . . . . 13 ⊢ ((𝐺:𝑇–1-1→𝒫 1o ∧ (⟨1o, ∅⟩ ∈ 𝑇 ∧ ⟨2o, ∅⟩ ∈ 𝑇)) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩) → ⟨1o, ∅⟩ = ⟨2o, ∅⟩))
76	1, 17, 52, 75	syl12anc 1197	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩) → ⟨1o, ∅⟩ = ⟨2o, ∅⟩))
77	74, 76	mtoi 636	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
78	77	ad2antrr 477	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
79		simpr 109	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → (𝐺‘⟨1o, ∅⟩) = {∅})
80	79	eqeq1d 2123	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩) ↔ {∅} = (𝐺‘⟨2o, ∅⟩)))
81	78, 80	mtbid 644	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ {∅} = (𝐺‘⟨2o, ∅⟩))
82		eqcom 2117	. . . . . . . . 9 ⊢ ({∅} = (𝐺‘⟨2o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = {∅})
83	81, 82	sylnib 648	. . . . . . . 8 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (𝐺‘⟨2o, ∅⟩) = {∅})
84	61, 83	jca 302	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
85	3, 52	ffvelrnd 5510	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨2o, ∅⟩) ∈ 𝒫 1o)
86	85	elpwid 3487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨2o, ∅⟩) ⊆ 1o)
87	86, 20	syl6sseq 3111	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨2o, ∅⟩) ⊆ {∅})
88		pwtrufal 12884	. . . . . . . . . 10 ⊢ ((𝐺‘⟨2o, ∅⟩) ⊆ {∅} → ¬ ¬ ((𝐺‘⟨2o, ∅⟩) = ∅ ∨ (𝐺‘⟨2o, ∅⟩) = {∅}))
89	87, 88	syl 14	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ ¬ ((𝐺‘⟨2o, ∅⟩) = ∅ ∨ (𝐺‘⟨2o, ∅⟩) = {∅}))
90		ioran 724	. . . . . . . . 9 ⊢ (¬ ((𝐺‘⟨2o, ∅⟩) = ∅ ∨ (𝐺‘⟨2o, ∅⟩) = {∅}) ↔ (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
91	89, 90	sylnib 648	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
92	91	ad2antrr 477	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
93	84, 92	pm2.65da 633	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → ¬ (𝐺‘⟨1o, ∅⟩) = {∅})
94	45, 93	jca 302	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → (¬ (𝐺‘⟨1o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨1o, ∅⟩) = {∅}))
95	25, 94	mtand 637	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨∅, ∅⟩) = ∅)
96		eqcom 2117	. . . . . . . . . . 11 ⊢ ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨1o, ∅⟩))
97	77, 96	sylnib 648	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨1o, ∅⟩))
98	97	ad2antrr 477	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨1o, ∅⟩))
99		simpr 109	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → (𝐺‘⟨1o, ∅⟩) = ∅)
100	99	eqeq2d 2126	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ((𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨1o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = ∅))
101	98, 100	mtbid 644	. . . . . . . 8 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨2o, ∅⟩) = ∅)
102	55	ad2antrr 477	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
103		eqcom 2117	. . . . . . . . . 10 ⊢ ((𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
104	102, 103	sylnib 648	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
105		simplr 502	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → (𝐺‘⟨∅, ∅⟩) = {∅})
106	105	eqeq2d 2126	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ((𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = {∅}))
107	104, 106	mtbid 644	. . . . . . . 8 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨2o, ∅⟩) = {∅})
108	101, 107	jca 302	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
109	91	ad2antrr 477	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
110	108, 109	pm2.65da 633	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → ¬ (𝐺‘⟨1o, ∅⟩) = ∅)
111	41	adantr 272	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
112		simpr 109	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → (𝐺‘⟨∅, ∅⟩) = {∅})
113	112	eqeq2d 2126	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) ↔ (𝐺‘⟨1o, ∅⟩) = {∅}))
114	111, 113	mtbid 644	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → ¬ (𝐺‘⟨1o, ∅⟩) = {∅})
115	110, 114	jca 302	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → (¬ (𝐺‘⟨1o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨1o, ∅⟩) = {∅}))
116	25, 115	mtand 637	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨∅, ∅⟩) = {∅})
117	95, 116	jca 302	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (¬ (𝐺‘⟨∅, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨∅, ∅⟩) = {∅}))
118	3, 38	ffvelrnd 5510	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨∅, ∅⟩) ∈ 𝒫 1o)
119	118	elpwid 3487	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨∅, ∅⟩) ⊆ 1o)
120	119, 20	syl6sseq 3111	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨∅, ∅⟩) ⊆ {∅})
121		pwtrufal 12884	. . . . 5 ⊢ ((𝐺‘⟨∅, ∅⟩) ⊆ {∅} → ¬ ¬ ((𝐺‘⟨∅, ∅⟩) = ∅ ∨ (𝐺‘⟨∅, ∅⟩) = {∅}))
122	120, 121	syl 14	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ ¬ ((𝐺‘⟨∅, ∅⟩) = ∅ ∨ (𝐺‘⟨∅, ∅⟩) = {∅}))
123		ioran 724	. . . 4 ⊢ (¬ ((𝐺‘⟨∅, ∅⟩) = ∅ ∨ (𝐺‘⟨∅, ∅⟩) = {∅}) ↔ (¬ (𝐺‘⟨∅, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨∅, ∅⟩) = {∅}))
124	122, 123	sylnib 648	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (¬ (𝐺‘⟨∅, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨∅, ∅⟩) = {∅}))
125	117, 124	pm2.65da 633	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → ¬ 2o ∈ 𝑁)
126		2onn 6371	. . . 4 ⊢ 2o ∈ ω
127		nntri1 6346	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 2o ∈ ω) → (𝑁 ⊆ 2o ↔ ¬ 2o ∈ 𝑁))
128	126, 127	mpan2 419	. . 3 ⊢ (𝑁 ∈ ω → (𝑁 ⊆ 2o ↔ ¬ 2o ∈ 𝑁))
129	128	adantr 272	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → (𝑁 ⊆ 2o ↔ ¬ 2o ∈ 𝑁))
130	125, 129	mpbird 166	1 ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → 𝑁 ⊆ 2o)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 680   = wceq 1314   ∈ wcel 1463   ⊆ wss 3037  ∅c0 3329  𝒫 cpw 3476  {csn 3493  ⟨cop 3496  ∪ ciun 3779  Oncon0 4245  suc csuc 4247  ωcom 4464   × cxp 4497  ⟶wf 5077  –1-1→wf1 5078  ‘cfv 5081  1_oc1o 6260  2_oc2o 6261

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fv 5089  df-1o 6267  df-2o 6268

This theorem is referenced by:  pwf1oexmid  12886