Theorem prconf1acwevd 35960

Description: If 𝐹 maps the proper class 𝑊 one-to-one into the ordinals and the Axiom of Choice holds, then 𝑅 well-orders the universe. (Contributed by BTernaryTau, 16-Sep-2026.)

Hypotheses

Ref	Expression
prconf1acwevd.1	⊢ (𝜑 → (¬ 𝑊 ∈ V → 𝐹:𝑊–1-1→On))
prconf1acwevd.2	⊢ (𝜑 → CHOICE)
prconf1acwevd.3	⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}
prconf1acwevd.4	⊢ 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}
prconf1acwevd.5	⊢ 𝑆 = (◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})

Assertion

Ref	Expression
prconf1acwevd	⊢ (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 prconf1acwevd

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

Step	Hyp	Ref	Expression
1		prconf1acwevd.4	. 2 ⊢ 𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}
2		prconf1acwevd.5	. 2 ⊢ 𝑆 = (◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})
3		prconf1acwevd.2	. . . . . . 7 ⊢ (𝜑 → CHOICE)
4		prconf1acwevd.3	. . . . . . . 8 ⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}
5	4	acwer1prc 35809	. . . . . . 7 ⊢ (CHOICE → ¬ 𝑊 ∈ V)
6	3, 5	syl 18	. . . . . 6 ⊢ (𝜑 → ¬ 𝑊 ∈ V)
7		prconf1acwevd.1	. . . . . 6 ⊢ (𝜑 → (¬ 𝑊 ∈ V → 𝐹:𝑊–1-1→On))
8	6, 7	mpd 16	. . . . 5 ⊢ (𝜑 → 𝐹:𝑊–1-1→On)
9		f1f 6778	. . . . 5 ⊢ (𝐹:𝑊–1-1→On → 𝐹:𝑊⟶On)
10		frn 6717	. . . . 5 ⊢ (𝐹:𝑊⟶On → ran 𝐹 ⊆ On)
11	8, 9, 10	3syl 19	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ On)
12	11	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ran 𝐹 ⊆ On)
13		ssid 3930	. . . . . . . . 9 ⊢ 𝑊 ⊆ 𝑊
14	4	onprcf1acwevdlem1 35957	. . . . . . . . 9 ⊢ ((𝑊 ⊆ 𝑊 ∧ 𝑣 ∈ On ∧ ∀𝑢 ∈ 𝑊 ¬ 𝑢 We (𝑅1‘𝑣)) → 𝑊 ∈ V)
15	13, 14	mp3an1 1477	. . . . . . . 8 ⊢ ((𝑣 ∈ On ∧ ∀𝑢 ∈ 𝑊 ¬ 𝑢 We (𝑅1‘𝑣)) → 𝑊 ∈ V)
16	15	ex 418	. . . . . . 7 ⊢ (𝑣 ∈ On → (∀𝑢 ∈ 𝑊 ¬ 𝑢 We (𝑅1‘𝑣) → 𝑊 ∈ V))
17	16, 6	nsyli 158	. . . . . 6 ⊢ (𝑣 ∈ On → (𝜑 → ¬ ∀𝑢 ∈ 𝑊 ¬ 𝑢 We (𝑅1‘𝑣)))
18	17	impcom 413	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ¬ ∀𝑢 ∈ 𝑊 ¬ 𝑢 We (𝑅1‘𝑣))
19		dfrex2 3089	. . . . 5 ⊢ (∃𝑢 ∈ 𝑊 𝑢 We (𝑅1‘𝑣) ↔ ¬ ∀𝑢 ∈ 𝑊 ¬ 𝑢 We (𝑅1‘𝑣))
20	18, 19	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ∃𝑢 ∈ 𝑊 𝑢 We (𝑅1‘𝑣))
21		f1f1orn 6836	. . . . . . . . 9 ⊢ (𝐹:𝑊–1-1→On → 𝐹:𝑊–1-1-onto→ran 𝐹)
22		f1ocnv 6837	. . . . . . . . 9 ⊢ (𝐹:𝑊–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝑊)
23	8, 21, 22	3syl 19	. . . . . . . 8 ⊢ (𝜑 → ◡𝐹:ran 𝐹–1-1-onto→𝑊)
24		f1ofo 6832	. . . . . . . 8 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝑊 → ◡𝐹:ran 𝐹–onto→𝑊)
25		forn 6799	. . . . . . . 8 ⊢ (◡𝐹:ran 𝐹–onto→𝑊 → ran ◡𝐹 = 𝑊)
26	23, 24, 25	3syl 19	. . . . . . 7 ⊢ (𝜑 → ran ◡𝐹 = 𝑊)
27	26	rexeqdv 3320	. . . . . 6 ⊢ (𝜑 → (∃𝑢 ∈ ran ◡𝐹𝑢 We (𝑅1‘𝑣) ↔ ∃𝑢 ∈ 𝑊 𝑢 We (𝑅1‘𝑣)))
28		f1ofn 6825	. . . . . . 7 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝑊 → ◡𝐹 Fn ran 𝐹)
29		weeq1 5637	. . . . . . . 8 ⊢ (𝑢 = (◡𝐹‘𝑤) → (𝑢 We (𝑅1‘𝑣) ↔ (◡𝐹‘𝑤) We (𝑅1‘𝑣)))
30	29	rexrn 7087	. . . . . . 7 ⊢ (◡𝐹 Fn ran 𝐹 → (∃𝑢 ∈ ran ◡𝐹𝑢 We (𝑅1‘𝑣) ↔ ∃𝑤 ∈ ran 𝐹(◡𝐹‘𝑤) We (𝑅1‘𝑣)))
31	23, 28, 30	3syl 19	. . . . . 6 ⊢ (𝜑 → (∃𝑢 ∈ ran ◡𝐹𝑢 We (𝑅1‘𝑣) ↔ ∃𝑤 ∈ ran 𝐹(◡𝐹‘𝑤) We (𝑅1‘𝑣)))
32	27, 31	bitr3d 284	. . . . 5 ⊢ (𝜑 → (∃𝑢 ∈ 𝑊 𝑢 We (𝑅1‘𝑣) ↔ ∃𝑤 ∈ ran 𝐹(◡𝐹‘𝑤) We (𝑅1‘𝑣)))
33	32	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → (∃𝑢 ∈ 𝑊 𝑢 We (𝑅1‘𝑣) ↔ ∃𝑤 ∈ ran 𝐹(◡𝐹‘𝑤) We (𝑅1‘𝑣)))
34	20, 33	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ∃𝑤 ∈ ran 𝐹(◡𝐹‘𝑤) We (𝑅1‘𝑣))
35		ssrexv 3978	. . 3 ⊢ (ran 𝐹 ⊆ On → (∃𝑤 ∈ ran 𝐹(◡𝐹‘𝑤) We (𝑅1‘𝑣) → ∃𝑤 ∈ On (◡𝐹‘𝑤) We (𝑅1‘𝑣)))
36	12, 34, 35	sylc 66	. 2 ⊢ ((𝜑 ∧ 𝑣 ∈ On) → ∃𝑤 ∈ On (◡𝐹‘𝑤) We (𝑅1‘𝑣))
37	1, 2, 36	onprcf1acwevdlem2 35958	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 3895  ∩ cint 4906   class class class wbr 5102  {copab 5166   We wwe 5602   × cxp 5648  ^◡ccnv 5649  ran crn 5651  Oncon0 6362  suc csuc 6364   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538  𝑅₁cr1 9767  rankcrnk 9768  CHOICEwac 10195

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 5326  ax-pr 5390  ax-un 7751  ax-reg 9587  ax-inf2 9643

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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rpss 7739  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-smo 8355  df-recs 8380  df-rdg 8419  df-seqom 8459  df-1o 8477  df-2o 8478  df-er 8718  df-map 8850  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-oi 9505  df-wdom 9560  df-r1 9769  df-rank 9770  df-dju 9983  df-card 10021  df-ac 10196  df-fin2 10365  df-fin4 10366  df-fin3 10367  df-fin5 10368  df-fin6 10369  df-fin7 10370

This theorem is used by:  onf1oracwevd  35964