Theorem onf1oracwevd 36052

Description: If 𝐹 maps the proper class 𝑊 one-to-one into the ordinals or 𝐻 maps the ordinals one-to-one into the proper class 𝑊, and if every set can either be mapped one-to-one into the ordinals or have the ordinals mapped one-to-one into it, then 𝑅 well-orders the universe. This is the ZFC version of (9 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the first two hypotheses would be replaced by something like (𝜑 → ∀𝑋(∃𝐹𝐹:𝑋⟶1-1On ∨ ∃𝐻𝐻:On⟶1-1𝑋)). However, we cannot quantify over classes, so we split this into the case where 𝑋 is a set and the case where 𝑋 is the proper class 𝑊 which together are sufficient for this proof. (Contributed by BTernaryTau, 16-Sep-2026.)

Hypotheses

Ref	Expression
onf1oracwevd.1	⊢ (𝜑 → (𝐹:𝑊⟶1-1On ∨ 𝐻:On⟶1-1𝑊))
onf1oracwevd.2	⊢ (𝜑 → ∀𝑣(∃𝑓 𝑓:𝑣⟶1-1On ∨ ∃ℎ ℎ:On⟶1-1𝑣))
onf1oracwevd.3	⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}
onf1oracwevd.4	⊢ 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝐴𝑧))}
onf1oracwevd.5	⊢ 𝐴 = (𝐼‘∩ {𝑤 ∈ On ∣ (𝐼‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
onf1oracwevd.6	⊢ 𝐼 = if(𝐹:𝑊⟶1-1On, ◡𝐹, 𝐻)

Assertion

Ref	Expression
onf1oracwevd	⊢ (𝜑 → 𝑅 We V)

Distinct variable groups:   𝜑,𝑦   𝑧,𝐴   𝑣,𝑓   𝑥,𝑟   𝑤,𝐹,𝑦,𝑧   𝑤,𝐼,𝑦,𝑧   𝑤,𝑊,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑧, 𝑤, 𝑣, 𝑓, ℎ, 𝑟)   𝐴(𝑥, 𝑦, 𝑤, 𝑣, 𝑓, ℎ, 𝑟)   𝑅(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑓, ℎ, 𝑟)   𝐹(𝑥, 𝑣, 𝑓, ℎ, 𝑟)   𝐻(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑓, ℎ, 𝑟)   𝐼(𝑥, 𝑣, 𝑓, ℎ, 𝑟)   𝑊(𝑥, 𝑣, 𝑓, ℎ, 𝑟)

Proof of Theorem onf1oracwevd

Step	Hyp	Ref	Expression
1		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑊⟶1-1On) → 𝐹:𝑊⟶1-1On)
2	1	a1d 26	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑊⟶1-1On) → (¬ 𝑊 ∈ V → 𝐹:𝑊⟶1-1On))
3		onf1oracwevd.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑣(∃𝑓 𝑓:𝑣⟶1-1On ∨ ∃ℎ ℎ:On⟶1-1𝑣))
4		onprc 7816	. . . . . . . . . 10 ⊢ ¬ On ∈ V
5		f1dmex 7993	. . . . . . . . . . 11 ⊢ ((ℎ:On⟶1-1𝑣 ∧ 𝑣 ∈ V) → On ∈ V)
6	5	elvd 3460	. . . . . . . . . 10 ⊢ (ℎ:On⟶1-1𝑣 → On ∈ V)
7	4, 6	mto 200	. . . . . . . . 9 ⊢ ¬ ℎ:On⟶1-1𝑣
8	7	nex 1833	. . . . . . . 8 ⊢ ¬ ∃ℎ ℎ:On⟶1-1𝑣
9		orel2 904	. . . . . . . 8 ⊢ (¬ ∃ℎ ℎ:On⟶1-1𝑣 → ((∃𝑓 𝑓:𝑣⟶1-1On ∨ ∃ℎ ℎ:On⟶1-1𝑣) → ∃𝑓 𝑓:𝑣⟶1-1On))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ ((∃𝑓 𝑓:𝑣⟶1-1On ∨ ∃ℎ ℎ:On⟶1-1𝑣) → ∃𝑓 𝑓:𝑣⟶1-1On)
11	3, 10	sylg 1856	. . . . . 6 ⊢ (𝜑 → ∀𝑣∃𝑓 𝑓:𝑣⟶1-1On)
12		dfac15 35845	. . . . . 6 ⊢ (CHOICE ↔ ∀𝑣∃𝑓 𝑓:𝑣⟶1-1On)
13	11, 12	sylibr 237	. . . . 5 ⊢ (𝜑 → CHOICE)
14	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑊⟶1-1On) → CHOICE)
15		onf1oracwevd.3	. . . 4 ⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}
16		eqid 2763	. . . 4 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))} = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))}
17		eqid 2763	. . . 4 ⊢ (◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}) = (◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
18	2, 14, 15, 16, 17	prconf1acwevd 36048	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑊⟶1-1On) → {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))} We V)
19		onf1oracwevd.4	. . . . . 6 ⊢ 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝐴𝑧))}
20		onf1oracwevd.5	. . . . . . . . . . 11 ⊢ 𝐴 = (𝐼‘∩ {𝑤 ∈ On ∣ (𝐼‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
21		onf1oracwevd.6	. . . . . . . . . . . . 13 ⊢ 𝐼 = if(𝐹:𝑊⟶1-1On, ◡𝐹, 𝐻)
22		iftrue 4491	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑊⟶1-1On → if(𝐹:𝑊⟶1-1On, ◡𝐹, 𝐻) = ◡𝐹)
23	21, 22	eqtrid 2810	. . . . . . . . . . . 12 ⊢ (𝐹:𝑊⟶1-1On → 𝐼 = ◡𝐹)
24	23	fveq1d 6910	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑊⟶1-1On → (𝐼‘𝑤) = (◡𝐹‘𝑤))
25		eqidd 2764	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑊⟶1-1On → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc (rank‘𝑦)))
26	24, 25	weeq12d 5644	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑊⟶1-1On → ((𝐼‘𝑤) We (𝑅1‘suc (rank‘𝑦)) ↔ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))))
27	26	rabbidv 3423	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑊⟶1-1On → {𝑤 ∈ On ∣ (𝐼‘𝑤) We (𝑅1‘suc (rank‘𝑦))} = {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
28	27	inteqd 4915	. . . . . . . . . . . 12 ⊢ (𝐹:𝑊⟶1-1On → ∩ {𝑤 ∈ On ∣ (𝐼‘𝑤) We (𝑅1‘suc (rank‘𝑦))} = ∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
29	23, 28	fveq12d 6915	. . . . . . . . . . 11 ⊢ (𝐹:𝑊⟶1-1On → (𝐼‘∩ {𝑤 ∈ On ∣ (𝐼‘𝑤) We (𝑅1‘suc (rank‘𝑦))}) = (◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}))
30	20, 29	eqtrid 2810	. . . . . . . . . 10 ⊢ (𝐹:𝑊⟶1-1On → 𝐴 = (◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))}))
31	30	breqd 5117	. . . . . . . . 9 ⊢ (𝐹:𝑊⟶1-1On → (𝑦𝐴𝑧 ↔ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))
32	31	anbi2d 642	. . . . . . . 8 ⊢ (𝐹:𝑊⟶1-1On → (((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝐴𝑧) ↔ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧)))
33	32	orbi2d 929	. . . . . . 7 ⊢ (𝐹:𝑊⟶1-1On → (((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝐴𝑧)) ↔ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))))
34	33	opabbidv 5174	. . . . . 6 ⊢ (𝐹:𝑊⟶1-1On → {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝐴𝑧))} = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))})
35	19, 34	eqtrid 2810	. . . . 5 ⊢ (𝐹:𝑊⟶1-1On → 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))})
36		eqidd 2764	. . . . 5 ⊢ (𝐹:𝑊⟶1-1On → V = V)
37	35, 36	weeq12d 5644	. . . 4 ⊢ (𝐹:𝑊⟶1-1On → (𝑅 We V ↔ {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))} We V))
38	37	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑊⟶1-1On) → (𝑅 We V ↔ {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦(◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})𝑧))} We V))
39	18, 38	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑊⟶1-1On) → 𝑅 We V)
40		onf1oracwevd.1	. . . . . 6 ⊢ (𝜑 → (𝐹:𝑊⟶1-1On ∨ 𝐻:On⟶1-1𝑊))
41	40	orcanai 1018	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹:𝑊⟶1-1On) → 𝐻:On⟶1-1𝑊)
42		iffalse 4494	. . . . . . . 8 ⊢ (¬ 𝐹:𝑊⟶1-1On → if(𝐹:𝑊⟶1-1On, ◡𝐹, 𝐻) = 𝐻)
43	21, 42	eqtrid 2810	. . . . . . 7 ⊢ (¬ 𝐹:𝑊⟶1-1On → 𝐼 = 𝐻)
44		f1eq1 6795	. . . . . . 7 ⊢ (𝐼 = 𝐻 → (𝐼:On⟶1-1𝑊 ↔ 𝐻:On⟶1-1𝑊))
45	43, 44	syl 18	. . . . . 6 ⊢ (¬ 𝐹:𝑊⟶1-1On → (𝐼:On⟶1-1𝑊 ↔ 𝐻:On⟶1-1𝑊))
46	45	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐹:𝑊⟶1-1On) → (𝐼:On⟶1-1𝑊 ↔ 𝐻:On⟶1-1𝑊))
47	41, 46	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹:𝑊⟶1-1On) → 𝐼:On⟶1-1𝑊)
48	47	a1d 26	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹:𝑊⟶1-1On) → (¬ 𝑊 ∈ V → 𝐼:On⟶1-1𝑊))
49	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹:𝑊⟶1-1On) → CHOICE)
50	48, 49, 15, 19, 20	onprcf1acwevd 36047	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹:𝑊⟶1-1On) → 𝑅 We V)
51	39, 50	pm2.61dan 825	1 ⊢ (𝜑 → 𝑅 We V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ifcif 4485  ∩ cint 4910   class class class wbr 5106  {copab 5170   We wwe 5607   × cxp 5653  ^◡ccnv 5654  Oncon0 6380  suc csuc 6382  ⟶1-1wf1 6555  ‘cfv 6558  𝑅₁cr1 9793  rankcrnk 9794  CHOICEwac 10221

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-reg 9613  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-rpss 7763  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-smo 8382  df-recs 8407  df-rdg 8446  df-seqom 8486  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-oi 9531  df-wdom 9586  df-r1 9795  df-rank 9796  df-dju 10009  df-card 10047  df-ac 10222  df-fin2 10391  df-fin4 10392  df-fin3 10393  df-fin5 10394  df-fin6 10395  df-fin7 10396

This theorem is used by: (None)