Theorem onprcf1acwevd 35998

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–1-1→𝑋)), 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–1-1→𝑊))
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 7796	. . . . . . 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 35828	. . . . . . . . . . . 12 ⊢ (CHOICE → ¬ 𝑊 ∈ V)
7	4, 6	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑊 ∈ V)
8		onprcf1acwevd.1	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑊 ∈ V → 𝐹:On–1-1→𝑊))
9	7, 8	mpd 16	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:On–1-1→𝑊)
10		f1f1orn 6841	. . . . . . . . . 10 ⊢ (𝐹:On–1-1→𝑊 → 𝐹:On–1-1-onto→ran 𝐹)
11		f1of1 6828	. . . . . . . . . 10 ⊢ (𝐹:On–1-1-onto→ran 𝐹 → 𝐹:On–1-1→ran 𝐹)
12	9, 10, 11	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:On–1-1→ran 𝐹)
13		f1dmex 7973	. . . . . . . . 9 ⊢ ((𝐹:On–1-1→ran 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 6783	. . . . . . . 8 ⊢ (𝐹:On–1-1→𝑊 → 𝐹:On⟶𝑊)
19	9, 18	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:On⟶𝑊)
20	19	frnd 6723	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑊)
21	5	onprcf1acwevdlem1 35996	. . . . . . 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 3089	. . . 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 6715	. . . . 5 ⊢ (𝜑 → 𝐹 Fn On)
28		weeq1 5638	. . . . . 6 ⊢ (𝑢 = (𝐹‘𝑤) → (𝑢 We (𝑅1‘𝑣) ↔ (𝐹‘𝑤) We (𝑅1‘𝑣)))
29	28	rexrn 7092	. . . . 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 35997	1 ⊢ (𝜑 → 𝑅 We V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896  ∩ cint 4906   class class class wbr 5102  {copab 5166   We wwe 5603   × cxp 5649  ran crn 5652  Oncon0 6365  suc csuc 6367   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543  𝑅₁cr1 9773  rankcrnk 9774  CHOICEwac 10201

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-reg 9593  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-rpss 7743  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-smo 8362  df-recs 8387  df-rdg 8426  df-seqom 8466  df-1o 8484  df-2o 8485  df-er 8725  df-map 8857  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-oi 9511  df-wdom 9566  df-r1 9775  df-rank 9776  df-dju 9989  df-card 10027  df-ac 10202  df-fin2 10371  df-fin4 10372  df-fin3 10373  df-fin5 10374  df-fin6 10375  df-fin7 10376

This theorem is used by:  onf1oracwevd  36003