Theorem subctctexmid 16277

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 4662	. . . . . . . 8 ⊢ ω ∈ V
3	2	rabex 4207	. . . . . . 7 ⊢ {𝑧 ∈ ω ∣ 𝑦 = {∅}} ∈ V
4	3	a1i 9	. . . . . 6 ⊢ (𝜑 → {𝑧 ∈ ω ∣ 𝑦 = {∅}} ∈ V)
5		ssrab2 3289	. . . . . . 7 ⊢ {𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊆ ω
6		f1oi 5587	. . . . . . . . 9 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}):{𝑧 ∈ ω ∣ 𝑦 = {∅}}–1-1-onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}
7		f1ofo 5555	. . . . . . . . 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 5026	. . . . . . . . . 10 ⊢ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ∈ V → ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}) ∈ V)
10	3, 9	ax-mp 5	. . . . . . . . 9 ⊢ ( I ↾ {𝑧 ∈ ω ∣ 𝑦 = {∅}}) ∈ V
11		foeq1 5520	. . . . . . . . 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 2878	. . . . . . . 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 3227	. . . . . . . 8 ⊢ (𝑠 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑠 ⊆ ω ↔ {𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊆ ω))
16		foeq2 5521	. . . . . . . . 9 ⊢ (𝑠 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}} ↔ 𝑓:{𝑧 ∈ ω ∣ 𝑦 = {∅}}–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
17	16	exbidv 1851	. . . . . . . 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 2871	. . . . . 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 1469	. . . . 5 ⊢ (𝜑 → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
21		foeq3 5522	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑓:𝑠–onto→𝑥 ↔ 𝑓:𝑠–onto→{𝑧 ∈ ω ∣ 𝑦 = {∅}}))
22	21	exbidv 1851	. . . . . . . . 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 1851	. . . . . . 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 7175	. . . . . . . . 9 ⊢ (𝑥 = {𝑧 ∈ ω ∣ 𝑦 = {∅}} → (𝑥 ⊔ 1o) = ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
26		foeq3 5522	. . . . . . . . 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 1851	. . . . . . 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 2877	. . . . 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 5602	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) → (ℎ‘𝑛) = ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛))
33	32	eqeq1d 2218	. . . . . . . . . . 11 ⊢ (ℎ = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) → ((ℎ‘𝑛) = 1o ↔ ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o))
34	33	rexbidv 2511	. . . . . . . . . 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 671	. . . . . . . . 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 671	. . . . . . . 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 7290	. . . . . . . . . 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 6558	. . . . . . . . . . . 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 6557	. . . . . . . . . . . 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 5524	. . . . . . . . . . . . . . 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 5744	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → (𝑔‘𝑛) ∈ ({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o))
52		eldju1st 7206	. . . . . . . . . . . . 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 6548	. . . . . . . . . . . . . . . 16 ⊢ 1o ≠ ∅
55	54	neii 2382	. . . . . . . . . . . . . . 15 ⊢ ¬ 1o = ∅
56		eqeq1 2216	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝑔‘𝑛)) = 1o → ((1st ‘(𝑔‘𝑛)) = ∅ ↔ 1o = ∅))
57	55, 56	mtbiri 679	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝑔‘𝑛)) = 1o → ¬ (1st ‘(𝑔‘𝑛)) = ∅)
58	57	orim2i 765	. . . . . . . . . . . . 13 ⊢ (((1st ‘(𝑔‘𝑛)) = ∅ ∨ (1st ‘(𝑔‘𝑛)) = 1o) → ((1st ‘(𝑔‘𝑛)) = ∅ ∨ ¬ (1st ‘(𝑔‘𝑛)) = ∅))
59		df-dc 839	. . . . . . . . . . . . 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 3612	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑛 ∈ ω) → if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) ∈ 2o)
63	62	fmpttd 5763	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (𝑛 ∈ ω ↦ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅)):ω⟶2o)
64		2fveq3 5608	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑛 → (1st ‘(𝑔‘𝑤)) = (1st ‘(𝑔‘𝑛)))
65	64	eqeq1d 2218	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑛 → ((1st ‘(𝑔‘𝑤)) = ∅ ↔ (1st ‘(𝑔‘𝑛)) = ∅))
66	65	ifbid 3604	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑛 → if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅) = if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅))
67		eqcom 2211	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑛 ↔ 𝑛 = 𝑤)
68		eqcom 2211	. . . . . . . . . . . 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 4159	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω ↦ if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅)) = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))
71	70	feq1i 5442	. . . . . . . . 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 6637	. . . . . . . . . 10 ⊢ 2o ∈ ω
74	73	elexi 2792	. . . . . . . . 9 ⊢ 2o ∈ V
75	74, 2	elmap 6794	. . . . . . . 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 2892	. . . . . 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 2209	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅)) = (𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))
79	78, 66, 50, 62	fvmptd3 5701	. . . . . . . . . . . 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 2218	. . . . . . . . . . 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 2215	. . . . . . . . . . . . . . . 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 3619	. . . . . . . . . . . . . . . . . . 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 2209	. . . . . . . . . . . . . . . . . . 19 ⊢ 1o = 1o
87		orcom 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o) ∨ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅)) ↔ ((¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅) ∨ ((1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = 1o)))
88	55	intnan 933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ (¬ (1st ‘(𝑔‘𝑛)) = ∅ ∧ 1o = ∅)
89		biorf 748	. . . . . . . . . . . . . . . . . . . . 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 946	. . . . . . . . . . . . . . . . . 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 7207	. . . . . . . . . . . . . . 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 2939	. . . . . . . . . . . . . 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 2628	. . . . . . . . 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 4663	. . . . . . . . . . . . . . 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 2941	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) ∧ 𝑦 = {∅}) → ∅ ∈ {𝑧 ∈ ω ∣ 𝑦 = {∅}})
111		djulcl 7186	. . . . . . . . . . . . 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 5849	. . . . . . . . . . . 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 5603	. . . . . . . . . . . . . . . 16 ⊢ ((inl‘∅) = (𝑔‘𝑛) → (1st ‘(inl‘∅)) = (1st ‘(𝑔‘𝑛)))
117		1stinl 7209	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ ω → (1st ‘(inl‘∅)) = ∅)
118	107, 117	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘(inl‘∅)) = ∅
119	116, 118	eqtr3di 2257	. . . . . . . . . . . . . . 15 ⊢ ((inl‘∅) = (𝑔‘𝑛) → (1st ‘(𝑔‘𝑛)) = ∅)
120	119	iftrued 3589	. . . . . . . . . . . . . 14 ⊢ ((inl‘∅) = (𝑔‘𝑛) → if((1st ‘(𝑔‘𝑛)) = ∅, 1o, ∅) = 1o)
121	115, 120	sylan9eq 2262	. . . . . . . . . . . . 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 2612	. . . . . . . . . . 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 671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔:ω–onto→({𝑧 ∈ ω ∣ 𝑦 = {∅}} ⊔ 1o)) → (¬ ∃𝑛 ∈ ω ((𝑤 ∈ ω ↦ if((1st ‘(𝑔‘𝑤)) = ∅, 1o, ∅))‘𝑛) = 1o ↔ ¬ 𝑦 = {∅}))
128	127	notbid 671	. . . . . 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 835	. . . . 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 1925	. . 3 ⊢ (𝜑 → STAB 𝑦 = {∅})
133	132	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝑦 ⊆ {∅}) → STAB 𝑦 = {∅})
134	133	exmid1stab 4271	1 ⊢ (𝜑 → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 712  STAB wstab 834  DECID wdc 838  ∀wal 1373   = wceq 1375  ∃wex 1518   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489  {crab 2492  Vcvv 2779   ⊆ wss 3177  ∅c0 3471  ifcif 3582  {csn 3646   ↦ cmpt 4124  EXMIDwem 4257   I cid 4356  ωcom 4659   ↾ cres 4698  ⟶wf 5290  –onto→wfo 5292  –1-1-onto→wf1o 5293  ‘cfv 5294  (class class class)co 5974  1^st c1st 6254  2^nd c2nd 6255  1_oc1o 6525  2_oc2o 6526   ↑_𝑚 cmap 6765   ⊔ cdju 7172  inlcinl 7180  Markovcmarkov 7286

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-exmid 4258  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-1o 6532  df-2o 6533  df-map 6767  df-dju 7173  df-inl 7182  df-inr 7183  df-markov 7287

This theorem is referenced by: (None)