Theorem onprcf1acwevd 36088

Description: If 𝐹 maps the ordinals one-to-one into the proper class 𝑊 and the Axiom of Choice holds, then 𝑅 well-orders the universe. This is the ZFC version of (7 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the first hypothesis would be something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On⟶1-1𝑋)), but since we cannot quantify over classes, we instead consider only the case 𝑋 = 𝑊 which is sufficient for this proof. (Contributed by BTernaryTau, 16-Sep-2026.)

Hypotheses

Ref	Expression
onprcf1acwevd.1	⊢ (𝜑 → (¬ 𝑊 ∈ V → 𝐹:On⟶1-1𝑊))
onprcf1acwevd.2	⊢ (𝜑 → CHOICE)
onprcf1acwevd.3	⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}
onprcf1acwevd.4	⊢ 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}
onprcf1acwevd.5	⊢ 𝑆 = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})

Assertion

Ref	Expression
onprcf1acwevd	⊢ (𝜑 → 𝑅 We V)

Distinct variable groups:   𝑧,𝑆   𝜑,𝑦   𝑥,𝑟   𝑤,𝐹,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑧, 𝑤, 𝑟)   𝑅(𝑥, 𝑦, 𝑧, 𝑤, 𝑟)   𝑆(𝑥, 𝑦, 𝑤, 𝑟)   𝐹(𝑥, 𝑟)   𝑊(𝑥, 𝑦, 𝑧, 𝑤, 𝑟)

Proof of Theorem onprcf1acwevd

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onprcf1acwevd.4	. 2 ⊢ 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}
2		onprcf1acwevd.5	. 2 ⊢ 𝑆 = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
3		onprc 7824	. . . . . . 7 ⊢ ¬ On ∈ V
4		onprcf1acwevd.2	. . . . . . . . . . . 12 ⊢ (𝜑 → CHOICE)
5		onprcf1acwevd.3	. . . . . . . . . . . . 13 ⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}
6	5	acwer1prc 35893	. . . . . . . . . . . 12 ⊢ (CHOICE → ¬ 𝑊 ∈ V)
7	4, 6	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑊 ∈ V)
8		onprcf1acwevd.1	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑊 ∈ V → 𝐹:On⟶1-1𝑊))
9	7, 8	mpd 16	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:On⟶1-1𝑊)
10		f1f1orn 6866	. . . . . . . . . 10 ⊢ (𝐹:On⟶1-1𝑊 → 𝐹:On⟶1-1ontoran 𝐹)
11		f1of1 6853	. . . . . . . . . 10 ⊢ (𝐹:On⟶1-1ontoran 𝐹 → 𝐹:On⟶1-1ran 𝐹)
12	9, 10, 11	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:On⟶1-1ran 𝐹)
13		f1dmex 8003	. . . . . . . . 9 ⊢ ((𝐹:On⟶1-1ran 𝐹 ∧ ran 𝐹 ∈ V) → On ∈ V)
14	12, 13	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ∈ V) → On ∈ V)
15	14	ex 418	. . . . . . 7 ⊢ (𝜑 → (ran 𝐹 ∈ V → On ∈ V))
16	3, 15	mtoi 202	. . . . . 6 ⊢ (𝜑 → ¬ ran 𝐹 ∈ V)
17	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ¬ ran 𝐹 ∈ V)
18		f1f 6808	. . . . . . . 8 ⊢ (𝐹:On⟶1-1𝑊 → 𝐹:On⟶𝑊)
19	9, 18	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:On⟶𝑊)
20	19	frnd 6748	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑊)
21	5	onprcf1acwevdlem1 36086	. . . . . . 7 ⊢ ((ran 𝐹 ⊆ 𝑊 ∧ 𝑣 ∈ On ∧ ∀𝑢 ∈ ran 𝐹 ¬ 𝑢 We (𝑅1‘𝑣)) → ran 𝐹 ∈ V)
22	21	3expia 1139	. . . . . 6 ⊢ ((ran 𝐹 ⊆ 𝑊 ∧ 𝑣 ∈ On) → (∀𝑢 ∈ ran 𝐹 ¬ 𝑢 We (𝑅1‘𝑣) → ran 𝐹 ∈ V))
23	20, 22	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → (∀𝑢 ∈ ran 𝐹 ¬ 𝑢 We (𝑅1‘𝑣) → ran 𝐹 ∈ V))
24	17, 23	mtod 201	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ¬ ∀𝑢 ∈ ran 𝐹 ¬ 𝑢 We (𝑅1‘𝑣))
25		dfrex2 3093	. . . 4 ⊢ (∃𝑢 ∈ ran 𝐹 𝑢 We (𝑅1‘𝑣) ↔ ¬ ∀𝑢 ∈ ran 𝐹 ¬ 𝑢 We (𝑅1‘𝑣))
26	24, 25	sylibr 237	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ∃𝑢 ∈ ran 𝐹 𝑢 We (𝑅1‘𝑣))
27	19	ffnd 6740	. . . . 5 ⊢ (𝜑 → 𝐹 Fn On)
28		weeq1 5645	. . . . . 6 ⊢ (𝑢 = (𝐹‘𝑤) → (𝑢 We (𝑅1‘𝑣) ↔ (𝐹‘𝑤) We (𝑅1‘𝑣)))
29	28	rexrn 7119	. . . . 5 ⊢ (𝐹 Fn On → (∃𝑢 ∈ ran 𝐹 𝑢 We (𝑅1‘𝑣) ↔ ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑣)))
30	27, 29	syl 18	. . . 4 ⊢ (𝜑 → (∃𝑢 ∈ ran 𝐹 𝑢 We (𝑅1‘𝑣) ↔ ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑣)))
31	30	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → (∃𝑢 ∈ ran 𝐹 𝑢 We (𝑅1‘𝑣) ↔ ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑣)))
32	26, 31	mpbid 235	. 2 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑣))
33	1, 2, 32	onprcf1acwevdlem2 36087	1 ⊢ (𝜑 → 𝑅 We V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∩ cint 4912   class class class wbr 5108  {copab 5173   We wwe 5610   × cxp 5656  ran crn 5659  Oncon0 6388  suc csuc 6390   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566  𝑅₁cr1 9803  rankcrnk 9804  CHOICEwac 10233

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-reg 9623  ax-inf2 9679

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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-rpss 7771  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-smo 8392  df-recs 8417  df-rdg 8456  df-seqom 8496  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-oi 9541  df-wdom 9596  df-r1 9805  df-rank 9806  df-dju 10019  df-card 10057  df-ac 10234  df-fin2 10403  df-fin4 10404  df-fin3 10405  df-fin5 10406  df-fin6 10407  df-fin7 10408

This theorem is used by:  onf1oracwevd  36093