Theorem pwle2 15146

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 528	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → 𝐺:𝑇–1-1→𝒫 1o)
2		f1f 5436	. . . . . . . . . . 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 4624	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
5	4	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → 𝑁 ∈ On)
6		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → 2o ∈ 𝑁)
7		1lt2o 6461	. . . . . . . . . . . . . 14 ⊢ 1o ∈ 2o
8	6, 7	jctil 312	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (1o ∈ 2o ∧ 2o ∈ 𝑁))
9		ontr1 4404	. . . . . . . . . . . . 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 6459	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1o
12		opelxpi 4673	. . . . . . . . . . . 12 ⊢ ((1o ∈ 𝑁 ∧ ∅ ∈ 1o) → ⟨1o, ∅⟩ ∈ (𝑁 × 1o))
13	10, 11, 12	sylancl 413	. . . . . . . . . . 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 4701	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o) = (𝑁 × 1o)
16	14, 15	eqtri 2210	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑁 × 1o)
17	13, 16	eleqtrrdi 2283	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨1o, ∅⟩ ∈ 𝑇)
18	3, 17	ffvelcdmd 5668	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨1o, ∅⟩) ∈ 𝒫 1o)
19	18	elpwid 3601	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨1o, ∅⟩) ⊆ 1o)
20		df1o2 6448	. . . . . . . 8 ⊢ 1o = {∅}
21	19, 20	sseqtrdi 3218	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨1o, ∅⟩) ⊆ {∅})
22		pwtrufal 15145	. . . . . . 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 753	. . . . . 6 ⊢ (¬ ((𝐺‘⟨1o, ∅⟩) = ∅ ∨ (𝐺‘⟨1o, ∅⟩) = {∅}) ↔ (¬ (𝐺‘⟨1o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨1o, ∅⟩) = {∅}))
25	23, 24	sylnib 677	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (¬ (𝐺‘⟨1o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨1o, ∅⟩) = {∅}))
26		1n0 6451	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
27	26	neii 2362	. . . . . . . . . 10 ⊢ ¬ 1o = ∅
28		1oex 6443	. . . . . . . . . . 11 ⊢ 1o ∈ V
29		0ex 4145	. . . . . . . . . . 11 ⊢ ∅ ∈ V
30	28, 29	opth1 4251	. . . . . . . . . 10 ⊢ (⟨1o, ∅⟩ = ⟨∅, ∅⟩ → 1o = ∅)
31	27, 30	mto 663	. . . . . . . . 9 ⊢ ¬ ⟨1o, ∅⟩ = ⟨∅, ∅⟩
32		0lt2o 6460	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ 2o
33	6, 32	jctil 312	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (∅ ∈ 2o ∧ 2o ∈ 𝑁))
34		ontr1 4404	. . . . . . . . . . . . 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 4673	. . . . . . . . . . . 12 ⊢ ((∅ ∈ 𝑁 ∧ ∅ ∈ 1o) → ⟨∅, ∅⟩ ∈ (𝑁 × 1o))
37	35, 11, 36	sylancl 413	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨∅, ∅⟩ ∈ (𝑁 × 1o))
38	37, 16	eleqtrrdi 2283	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨∅, ∅⟩ ∈ 𝑇)
39		f1veqaeq 5786	. . . . . . . . . 10 ⊢ ((𝐺:𝑇–1-1→𝒫 1o ∧ (⟨1o, ∅⟩ ∈ 𝑇 ∧ ⟨∅, ∅⟩ ∈ 𝑇)) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) → ⟨1o, ∅⟩ = ⟨∅, ∅⟩))
40	1, 17, 38, 39	syl12anc 1247	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) → ⟨1o, ∅⟩ = ⟨∅, ∅⟩))
41	31, 40	mtoi 665	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
42	41	adantr 276	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
43		simpr 110	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → (𝐺‘⟨∅, ∅⟩) = ∅)
44	43	eqeq2d 2201	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) ↔ (𝐺‘⟨1o, ∅⟩) = ∅))
45	42, 44	mtbid 673	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → ¬ (𝐺‘⟨1o, ∅⟩) = ∅)
46		2on0 6445	. . . . . . . . . . . . . 14 ⊢ 2o ≠ ∅
47	46	nesymi 2406	. . . . . . . . . . . . 13 ⊢ ¬ ∅ = 2o
48	29, 29	opth1 4251	. . . . . . . . . . . . 13 ⊢ (⟨∅, ∅⟩ = ⟨2o, ∅⟩ → ∅ = 2o)
49	47, 48	mto 663	. . . . . . . . . . . 12 ⊢ ¬ ⟨∅, ∅⟩ = ⟨2o, ∅⟩
50		opelxpi 4673	. . . . . . . . . . . . . . 15 ⊢ ((2o ∈ 𝑁 ∧ ∅ ∈ 1o) → ⟨2o, ∅⟩ ∈ (𝑁 × 1o))
51	6, 11, 50	sylancl 413	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨2o, ∅⟩ ∈ (𝑁 × 1o))
52	51, 16	eleqtrrdi 2283	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ⟨2o, ∅⟩ ∈ 𝑇)
53		f1veqaeq 5786	. . . . . . . . . . . . 13 ⊢ ((𝐺:𝑇–1-1→𝒫 1o ∧ (⟨∅, ∅⟩ ∈ 𝑇 ∧ ⟨2o, ∅⟩ ∈ 𝑇)) → ((𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩) → ⟨∅, ∅⟩ = ⟨2o, ∅⟩))
54	1, 38, 52, 53	syl12anc 1247	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ((𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩) → ⟨∅, ∅⟩ = ⟨2o, ∅⟩))
55	49, 54	mtoi 665	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
56	55	ad2antrr 488	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
57		simplr 528	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → (𝐺‘⟨∅, ∅⟩) = ∅)
58	57	eqeq1d 2198	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ((𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩) ↔ ∅ = (𝐺‘⟨2o, ∅⟩)))
59	56, 58	mtbid 673	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ ∅ = (𝐺‘⟨2o, ∅⟩))
60		eqcom 2191	. . . . . . . . 9 ⊢ (∅ = (𝐺‘⟨2o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = ∅)
61	59, 60	sylnib 677	. . . . . . . 8 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (𝐺‘⟨2o, ∅⟩) = ∅)
62		1onn 6539	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ ω
63		peano1 4608	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ ω
64		peano4 4611	. . . . . . . . . . . . . . . . . 18 ⊢ ((1o ∈ ω ∧ ∅ ∈ ω) → (suc 1o = suc ∅ ↔ 1o = ∅))
65	62, 63, 64	mp2an 426	. . . . . . . . . . . . . . . . 17 ⊢ (suc 1o = suc ∅ ↔ 1o = ∅)
66	26, 65	nemtbir 2449	. . . . . . . . . . . . . . . 16 ⊢ ¬ suc 1o = suc ∅
67		df-2o 6436	. . . . . . . . . . . . . . . . 17 ⊢ 2o = suc 1o
68		df-1o 6435	. . . . . . . . . . . . . . . . 17 ⊢ 1o = suc ∅
69	67, 68	eqeq12i 2203	. . . . . . . . . . . . . . . 16 ⊢ (2o = 1o ↔ suc 1o = suc ∅)
70	66, 69	mtbir 672	. . . . . . . . . . . . . . 15 ⊢ ¬ 2o = 1o
71	70	neir 2363	. . . . . . . . . . . . . 14 ⊢ 2o ≠ 1o
72	71	nesymi 2406	. . . . . . . . . . . . 13 ⊢ ¬ 1o = 2o
73	28, 29	opth1 4251	. . . . . . . . . . . . 13 ⊢ (⟨1o, ∅⟩ = ⟨2o, ∅⟩ → 1o = 2o)
74	72, 73	mto 663	. . . . . . . . . . . 12 ⊢ ¬ ⟨1o, ∅⟩ = ⟨2o, ∅⟩
75		f1veqaeq 5786	. . . . . . . . . . . . 13 ⊢ ((𝐺:𝑇–1-1→𝒫 1o ∧ (⟨1o, ∅⟩ ∈ 𝑇 ∧ ⟨2o, ∅⟩ ∈ 𝑇)) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩) → ⟨1o, ∅⟩ = ⟨2o, ∅⟩))
76	1, 17, 52, 75	syl12anc 1247	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩) → ⟨1o, ∅⟩ = ⟨2o, ∅⟩))
77	74, 76	mtoi 665	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
78	77	ad2antrr 488	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
79		simpr 110	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → (𝐺‘⟨1o, ∅⟩) = {∅})
80	79	eqeq1d 2198	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩) ↔ {∅} = (𝐺‘⟨2o, ∅⟩)))
81	78, 80	mtbid 673	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ {∅} = (𝐺‘⟨2o, ∅⟩))
82		eqcom 2191	. . . . . . . . 9 ⊢ ({∅} = (𝐺‘⟨2o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = {∅})
83	81, 82	sylnib 677	. . . . . . . 8 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (𝐺‘⟨2o, ∅⟩) = {∅})
84	61, 83	jca 306	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
85	3, 52	ffvelcdmd 5668	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨2o, ∅⟩) ∈ 𝒫 1o)
86	85	elpwid 3601	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨2o, ∅⟩) ⊆ 1o)
87	86, 20	sseqtrdi 3218	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨2o, ∅⟩) ⊆ {∅})
88		pwtrufal 15145	. . . . . . . . . 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 753	. . . . . . . . 9 ⊢ (¬ ((𝐺‘⟨2o, ∅⟩) = ∅ ∨ (𝐺‘⟨2o, ∅⟩) = {∅}) ↔ (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
91	89, 90	sylnib 677	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
92	91	ad2antrr 488	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) ∧ (𝐺‘⟨1o, ∅⟩) = {∅}) → ¬ (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
93	84, 92	pm2.65da 662	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → ¬ (𝐺‘⟨1o, ∅⟩) = {∅})
94	45, 93	jca 306	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = ∅) → (¬ (𝐺‘⟨1o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨1o, ∅⟩) = {∅}))
95	25, 94	mtand 666	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨∅, ∅⟩) = ∅)
96		eqcom 2191	. . . . . . . . . . 11 ⊢ ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨2o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨1o, ∅⟩))
97	77, 96	sylnib 677	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨1o, ∅⟩))
98	97	ad2antrr 488	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨1o, ∅⟩))
99		simpr 110	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → (𝐺‘⟨1o, ∅⟩) = ∅)
100	99	eqeq2d 2201	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ((𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨1o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = ∅))
101	98, 100	mtbid 673	. . . . . . . 8 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨2o, ∅⟩) = ∅)
102	55	ad2antrr 488	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩))
103		eqcom 2191	. . . . . . . . . 10 ⊢ ((𝐺‘⟨∅, ∅⟩) = (𝐺‘⟨2o, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
104	102, 103	sylnib 677	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
105		simplr 528	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → (𝐺‘⟨∅, ∅⟩) = {∅})
106	105	eqeq2d 2201	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ((𝐺‘⟨2o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) ↔ (𝐺‘⟨2o, ∅⟩) = {∅}))
107	104, 106	mtbid 673	. . . . . . . 8 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (𝐺‘⟨2o, ∅⟩) = {∅})
108	101, 107	jca 306	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
109	91	ad2antrr 488	. . . . . . 7 ⊢ (((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) ∧ (𝐺‘⟨1o, ∅⟩) = ∅) → ¬ (¬ (𝐺‘⟨2o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨2o, ∅⟩) = {∅}))
110	108, 109	pm2.65da 662	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → ¬ (𝐺‘⟨1o, ∅⟩) = ∅)
111	41	adantr 276	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → ¬ (𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩))
112		simpr 110	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → (𝐺‘⟨∅, ∅⟩) = {∅})
113	112	eqeq2d 2201	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → ((𝐺‘⟨1o, ∅⟩) = (𝐺‘⟨∅, ∅⟩) ↔ (𝐺‘⟨1o, ∅⟩) = {∅}))
114	111, 113	mtbid 673	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → ¬ (𝐺‘⟨1o, ∅⟩) = {∅})
115	110, 114	jca 306	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) ∧ (𝐺‘⟨∅, ∅⟩) = {∅}) → (¬ (𝐺‘⟨1o, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨1o, ∅⟩) = {∅}))
116	25, 115	mtand 666	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (𝐺‘⟨∅, ∅⟩) = {∅})
117	95, 116	jca 306	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (¬ (𝐺‘⟨∅, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨∅, ∅⟩) = {∅}))
118	3, 38	ffvelcdmd 5668	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨∅, ∅⟩) ∈ 𝒫 1o)
119	118	elpwid 3601	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨∅, ∅⟩) ⊆ 1o)
120	119, 20	sseqtrdi 3218	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → (𝐺‘⟨∅, ∅⟩) ⊆ {∅})
121		pwtrufal 15145	. . . . 5 ⊢ ((𝐺‘⟨∅, ∅⟩) ⊆ {∅} → ¬ ¬ ((𝐺‘⟨∅, ∅⟩) = ∅ ∨ (𝐺‘⟨∅, ∅⟩) = {∅}))
122	120, 121	syl 14	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ ¬ ((𝐺‘⟨∅, ∅⟩) = ∅ ∨ (𝐺‘⟨∅, ∅⟩) = {∅}))
123		ioran 753	. . . 4 ⊢ (¬ ((𝐺‘⟨∅, ∅⟩) = ∅ ∨ (𝐺‘⟨∅, ∅⟩) = {∅}) ↔ (¬ (𝐺‘⟨∅, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨∅, ∅⟩) = {∅}))
124	122, 123	sylnib 677	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) ∧ 2o ∈ 𝑁) → ¬ (¬ (𝐺‘⟨∅, ∅⟩) = ∅ ∧ ¬ (𝐺‘⟨∅, ∅⟩) = {∅}))
125	117, 124	pm2.65da 662	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → ¬ 2o ∈ 𝑁)
126		2onn 6540	. . . 4 ⊢ 2o ∈ ω
127		nntri1 6515	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 2o ∈ ω) → (𝑁 ⊆ 2o ↔ ¬ 2o ∈ 𝑁))
128	126, 127	mpan2 425	. . 3 ⊢ (𝑁 ∈ ω → (𝑁 ⊆ 2o ↔ ¬ 2o ∈ 𝑁))
129	128	adantr 276	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → (𝑁 ⊆ 2o ↔ ¬ 2o ∈ 𝑁))
130	125, 129	mpbird 167	1 ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → 𝑁 ⊆ 2o)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2160   ⊆ wss 3144  ∅c0 3437  𝒫 cpw 3590  {csn 3607  ⟨cop 3610  ∪ ciun 3901  Oncon0 4378  suc csuc 4380  ωcom 4604   × cxp 4639  ⟶wf 5227  –1-1→wf1 5228  ‘cfv 5231  1_oc1o 6428  2_oc2o 6429

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fv 5239  df-1o 6435  df-2o 6436

This theorem is referenced by:  pwf1oexmid  15147