Theorem subctctexmid 15645

Description: If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)

Hypotheses

Ref	Expression
subctctexmid.x	⊢ (𝜑 → ∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)))
subctctexmid.mk	⊢ (𝜑 → ω ∈ Markov)

Assertion

Ref	Expression
subctctexmid	⊢ (𝜑 → EXMID)

Distinct variable groups:   𝑓,𝑠,𝑥   𝜑,𝑔   𝑥,𝑔

Allowed substitution hints:   𝜑(𝑥,𝑓,𝑠)

Proof of Theorem subctctexmid

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

Step	Hyp	Ref	Expression
1		subctctexmid.x	. . . . 5 ⊢ (𝜑 → ∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)))
2		omex 4629	. . . . . . . 8 ⊢ ω ∈ V
3	2	rabex 4177	. . . . . . 7 ⊢ {𝑧 ∈ ω ∣ 𝑦 = {∅}} ∈ V
4	3	a1i 9	. . . . . 6 ⊢ (𝜑 → {𝑧 ∈ ω ∣ 𝑦 = {∅}} ∈ V)
5		ssrab2 3268	. . . . . . 7 ⊢ {𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊆ ω
6		f1oi 5542	. . . . . . . . 9 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}):{𝑧 ∈ ω ∣ 𝑦 = {∅}}–1-1-onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}
7		f1ofo 5511	. . . . . . . . 9 ⊢ (( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}):{𝑧 ∈ ω ∣ 𝑦 = {∅}}–1-1-onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}} → ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}):{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}})
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}):{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}
9		resiexg 4991	. . . . . . . . . 10 ⊢ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ∈ V → ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}) ∈ V)
10	3, 9	ax-mp 5	. . . . . . . . 9 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}) ∈ V
11		foeq1 5476	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}) → (𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}} ↔ ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}):{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
12	10, 11	spcev 2859	. . . . . . . 8 ⊢ (( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}):{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}} → ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}})
13	8, 12	ax-mp 5	. . . . . . 7 ⊢ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}
14	5, 13	pm3.2i 272	. . . . . 6 ⊢ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊆ ω ∧ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}})
15		sseq1 3206	. . . . . . . 8 ⊢ (𝑠 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑠 ⊆ ω ↔ {𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊆ ω))
16		foeq2 5477	. . . . . . . . 9 ⊢ (𝑠 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}} ↔ 𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
17	16	exbidv 1839	. . . . . . . 8 ⊢ (𝑠 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}} ↔ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
18	15, 17	anbi12d 473	. . . . . . 7 ⊢ (𝑠 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → ((𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}) ↔ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊆ ω ∧ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}})))
19	18	spcegv 2852	. . . . . 6 ⊢ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ∈ V → (({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊆ ω ∧ ∃𝑓 𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}})))
20	4, 14, 19	mpisyl 1457	. . . . 5 ⊢ (𝜑 → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
21		foeq3 5478	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑓:𝑠–onto→𝑥 ↔ 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
22	21	exbidv 1839	. . . . . . . . 9 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (∃𝑓 𝑓:𝑠–onto→𝑥 ↔ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
23	22	anbi2d 464	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → ((𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) ↔ (𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}})))
24	23	exbidv 1839	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) ↔ ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}})))
25		djueq1 7106	. . . . . . . . 9 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
26		foeq3 5478	. . . . . . . . 9 ⊢ ((𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o) → (𝑔:ω–onto→(𝑥 ⊔ 1o) ↔ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)))
27	25, 26	syl 14	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑔:ω–onto→(𝑥 ⊔ 1o) ↔ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)))
28	27	exbidv 1839	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)))
29	24, 28	imbi12d 234	. . . . . 6 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → ((∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)) ↔ (∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}) → ∃𝑔 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))))
30	3, 29	spcv 2858	. . . . 5 ⊢ (∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)) → (∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}) → ∃𝑔 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)))
31	1, 20, 30	sylc 62	. . . 4 ⊢ (𝜑 → ∃𝑔 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
32		fveq1 5557	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) → (ℎ‘𝑛) = ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛))
33	32	eqeq1d 2205	. . . . . . . . . . 11 ⊢ (ℎ = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) → ((ℎ‘𝑛) = 1o ↔ ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
34	33	rexbidv 2498	. . . . . . . . . 10 ⊢ (ℎ = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) → (∃𝑛 ∈ ω (ℎ‘𝑛) = 1o ↔ ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
35	34	notbid 668	. . . . . . . . 9 ⊢ (ℎ = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) → (¬ ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o ↔ ¬ ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
36	35	notbid 668	. . . . . . . 8 ⊢ (ℎ = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) → (¬ ¬ ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o ↔ ¬ ¬ ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
37	36, 34	imbi12d 234	. . . . . . 7 ⊢ (ℎ = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) → ((¬ ¬ ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o → ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o) ↔ (¬ ¬ ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o → ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o)))
38		subctctexmid.mk	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ Markov)
39		ismkvnex 7221	. . . . . . . . . 10 ⊢ (ω ∈ Markov → (ω ∈ Markov ↔ ∀ℎ ∈ (2o ↑𝑚 ω)(¬ ¬ ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o → ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o)))
40	38, 39	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (ω ∈ Markov ↔ ∀ℎ ∈ (2o ↑𝑚 ω)(¬ ¬ ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o → ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o)))
41	38, 40	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → ∀ℎ ∈ (2o ↑𝑚 ω)(¬ ¬ ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o → ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o))
42	41	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → ∀ℎ ∈ (2o ↑𝑚 ω)(¬ ¬ ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o → ∃𝑛 ∈ ω (ℎ‘𝑛) = 1o))
43		1lt2o 6500	. . . . . . . . . . . 12 ⊢ 1o ∈ 2o
44	43	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ (1st ‘(𝑔‘𝑛)) = ∅) → 1o ∈ 2o)
45		0lt2o 6499	. . . . . . . . . . . 12 ⊢ ∅ ∈ 2o
46	45	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ ¬ (1st ‘(𝑔‘𝑛)) = ∅) → ∅ ∈ 2o)
47		simplr 528	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
48		fof 5480	. . . . . . . . . . . . . . 15 ⊢ (𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o) → 𝑔:ω⟶({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
49	47, 48	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → 𝑔:ω⟶({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
50		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
51	49, 50	ffvelcdmd 5698	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → (𝑔‘𝑛) ∈ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
52		eldju1st 7137	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑛) ∈ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o) → ((1st ‘(𝑔‘𝑛)) = ∅ ∨ (1st ‘(𝑔‘𝑛)) = 1o))
53	51, 52	syl 14	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → ((1st ‘(𝑔‘𝑛)) = ∅ ∨ (1st ‘(𝑔‘𝑛)) = 1o))
54		1n0 6490	. . . . . . . . . . . . . . . 16 ⊢ 1o ≠ ∅
55	54	neii 2369	. . . . . . . . . . . . . . 15 ⊢ ¬ 1o = ∅
56		eqeq1 2203	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝑔‘𝑛)) = 1o → ((1st ‘(𝑔‘𝑛)) = ∅ ↔ 1o = ∅))
57	55, 56	mtbiri 676	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝑔‘𝑛)) = 1o → ¬ (1st ‘(𝑔‘𝑛)) = ∅)
58	57	orim2i 762	. . . . . . . . . . . . 13 ⊢ (((1st ‘(𝑔‘𝑛)) = ∅ ∨ (1st ‘(𝑔‘𝑛)) = 1o) → ((1st ‘(𝑔‘𝑛)) = ∅ ∨ ¬ (1st ‘(𝑔‘𝑛)) = ∅))
59		df-dc 836	. . . . . . . . . . . . 13 ⊢ (DECID (1st ‘(𝑔‘𝑛)) = ∅ ↔ ((1st ‘(𝑔‘𝑛)) = ∅ ∨ ¬ (1st ‘(𝑔‘𝑛)) = ∅))
60	58, 59	sylibr 134	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝑔‘𝑛)) = ∅ ∨ (1st ‘(𝑔‘𝑛)) = 1o) → DECID (1st ‘(𝑔‘𝑛)) = ∅)
61	53, 60	syl 14	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → DECID (1st ‘(𝑔‘𝑛)) = ∅)
62	44, 46, 61	ifcldadc 3590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) ∈ 2o)
63	62	fmpttd 5717	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (𝑛 ∈ ω ↦ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅)):ω⟶2o)
64		2fveq3 5563	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑛 → (1st ‘(𝑔‘𝑤)) = (1st ‘(𝑔‘𝑛)))
65	64	eqeq1d 2205	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑛 → ((1st ‘(𝑔‘𝑤)) = ∅ ↔ (1st ‘(𝑔‘𝑛)) = ∅))
66	65	ifbid 3582	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑛 → if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅) = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅))
67		eqcom 2198	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑛 ↔ 𝑛 = 𝑤)
68		eqcom 2198	. . . . . . . . . . . 12 ⊢ (if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅) = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) ↔ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))
69	66, 67, 68	3imtr3i 200	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑤 → if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))
70	69	cbvmptv 4129	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω ↦ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅)) = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))
71	70	feq1i 5400	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ↦ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅)):ω⟶2o ↔ (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)):ω⟶2o)
72	63, 71	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)):ω⟶2o)
73		2onn 6579	. . . . . . . . . 10 ⊢ 2o ∈ ω
74	73	elexi 2775	. . . . . . . . 9 ⊢ 2o ∈ V
75	74, 2	elmap 6736	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) ∈ (2o ↑𝑚 ω) ↔ (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)):ω⟶2o)
76	72, 75	sylibr 134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) ∈ (2o ↑𝑚 ω))
77	37, 42, 76	rspcdva 2873	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (¬ ¬ ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o → ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
78		eqid 2196	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))
79	78, 66, 50, 62	fvmptd3 5655	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅))
80	79	eqeq1d 2205	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → (((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o ↔ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o))
81	51	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o) → (𝑔‘𝑛) ∈ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
82		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o) → if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o)
83	82	eqcomd 2202	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o) → 1o = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅))
84		eqifdc 3596	. . . . . . . . . . . . . . . . . . 19 ⊢ (DECID (1st ‘(𝑔‘𝑛)) = ∅ → (1o = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) ↔ (((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o) ∨ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅))))
85	61, 84	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → (1o = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) ↔ (((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o) ∨ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅))))
86		eqid 2196	. . . . . . . . . . . . . . . . . . 19 ⊢ 1o = 1o
87		orcom 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o) ∨ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅)) ↔ ((¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅) ∨ ((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o)))
88	55	intnan 930	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅)
89		biorf 745	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅) → (((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o) ↔ ((¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅) ∨ ((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o))))
90	88, 89	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o) ↔ ((¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅) ∨ ((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o)))
91	87, 90	bitr4i 187	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o) ∨ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅)) ↔ ((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o))
92	86, 91	mpbiran2 943	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o) ∨ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅)) ↔ (1st ‘(𝑔‘𝑛)) = ∅)
93	85, 92	bitrdi 196	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → (1o = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) ↔ (1st ‘(𝑔‘𝑛)) = ∅))
94	93	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o) → (1o = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) ↔ (1st ‘(𝑔‘𝑛)) = ∅))
95	83, 94	mpbid 147	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o) → (1st ‘(𝑔‘𝑛)) = ∅)
96		eldju2ndl 7138	. . . . . . . . . . . . . . 15 ⊢ (((𝑔‘𝑛) ∈ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o) ∧ (1st ‘(𝑔‘𝑛)) = ∅) → (2nd ‘(𝑔‘𝑛)) ∈ {𝑧 ∈ ω ∣ 𝑦 = {∅}})
97	81, 95, 96	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o) → (2nd ‘(𝑔‘𝑛)) ∈ {𝑧 ∈ ω ∣ 𝑦 = {∅}})
98		biidd 172	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (2nd ‘(𝑔‘𝑛)) → (𝑦 = {∅} ↔ 𝑦 = {∅}))
99	98	elrab 2920	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘(𝑔‘𝑛)) ∈ {𝑧 ∈ ω ∣ 𝑦 = {∅}} ↔ ((2nd ‘(𝑔‘𝑛)) ∈ ω ∧ 𝑦 = {∅}))
100	97, 99	sylib 122	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o) → ((2nd ‘(𝑔‘𝑛)) ∈ ω ∧ 𝑦 = {∅}))
101	100	simprd 114	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) ∧ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o) → 𝑦 = {∅})
102	101	ex 115	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → (if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o → 𝑦 = {∅}))
103	80, 102	sylbid 150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → (((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o → 𝑦 = {∅}))
104	103	rexlimdva 2614	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o → 𝑦 = {∅}))
105		simplr 528	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
106		biidd 172	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝑦 = {∅} ↔ 𝑦 = {∅}))
107		peano1 4630	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
108	107	a1i 9	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → ∅ ∈ ω)
109		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → 𝑦 = {∅})
110	106, 108, 109	elrabd 2922	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → ∅ ∈ {𝑧 ∈ ω ∣ 𝑦 = {∅}})
111		djulcl 7117	. . . . . . . . . . . . 13 ⊢ (∅ ∈ {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (inl‘∅) ∈ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
112	110, 111	syl 14	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → (inl‘∅) ∈ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
113		foelrn 5799	. . . . . . . . . . . 12 ⊢ ((𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o) ∧ (inl‘∅) ∈ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → ∃𝑛 ∈ ω (inl‘∅) = (𝑔‘𝑛))
114	105, 112, 113	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → ∃𝑛 ∈ ω (inl‘∅) = (𝑔‘𝑛))
115	79	adantlr 477	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) ∧ 𝑛 ∈ ω) → ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅))
116		fveq2 5558	. . . . . . . . . . . . . . . 16 ⊢ ((inl‘∅) = (𝑔‘𝑛) → (1st ‘(inl‘∅)) = (1st ‘(𝑔‘𝑛)))
117		1stinl 7140	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ ω → (1st ‘(inl‘∅)) = ∅)
118	107, 117	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘(inl‘∅)) = ∅
119	116, 118	eqtr3di 2244	. . . . . . . . . . . . . . 15 ⊢ ((inl‘∅) = (𝑔‘𝑛) → (1st ‘(𝑔‘𝑛)) = ∅)
120	119	iftrued 3568	. . . . . . . . . . . . . 14 ⊢ ((inl‘∅) = (𝑔‘𝑛) → if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o)
121	115, 120	sylan9eq 2249	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) ∧ 𝑛 ∈ ω) ∧ (inl‘∅) = (𝑔‘𝑛)) → ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o)
122	121	ex 115	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) ∧ 𝑛 ∈ ω) → ((inl‘∅) = (𝑔‘𝑛) → ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
123	122	reximdva 2599	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → (∃𝑛 ∈ ω (inl‘∅) = (𝑔‘𝑛) → ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
124	114, 123	mpd 13	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o)
125	124	ex 115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (𝑦 = {∅} → ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
126	104, 125	impbid 129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o ↔ 𝑦 = {∅}))
127	126	notbid 668	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (¬ ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o ↔ ¬ 𝑦 = {∅}))
128	127	notbid 668	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (¬ ¬ ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o ↔ ¬ ¬ 𝑦 = {∅}))
129	77, 128, 126	3imtr3d 202	. . . . 5 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (¬ ¬ 𝑦 = {∅} → 𝑦 = {∅}))
130		df-stab 832	. . . . 5 ⊢ (STAB 𝑦 = {∅} ↔ (¬ ¬ 𝑦 = {∅} → 𝑦 = {∅}))
131	129, 130	sylibr 134	. . . 4 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → STAB 𝑦 = {∅})
132	31, 131	exlimddv 1913	. . 3 ⊢ (𝜑 → STAB 𝑦 = {∅})
133	132	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → STAB 𝑦 = {∅})
134	133	exmid1stab 4241	1 ⊢ (𝜑 → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  STAB wstab 831  DECID wdc 835  ∀wal 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479  Vcvv 2763   ⊆ wss 3157  ∅c0 3450  ifcif 3561  {csn 3622   ↦ cmpt 4094  EXMIDwem 4227   I cid 4323  ωcom 4626   ↾ cres 4665  ⟶wf 5254  –onto→wfo 5256  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922  1^st c1st 6196  2^nd c2nd 6197  1_oc1o 6467  2_oc2o 6468   ↑_𝑚 cmap 6707   ⊔ cdju 7103  inlcinl 7111  Markovcmarkov 7217

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-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-exmid 4228  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-1o 6474  df-2o 6475  df-map 6709  df-dju 7104  df-inl 7113  df-inr 7114  df-markov 7218

This theorem is referenced by: (None)