Theorem prconf1acwevd 36048

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⟶1-1On))
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 35852	. . . . . . 7 ⊢ (CHOICE → ¬ 𝑊 ∈ V)
6	3, 5	syl 18	. . . . . 6 ⊢ (𝜑 → ¬ 𝑊 ∈ V)
7		prconf1acwevd.1	. . . . . 6 ⊢ (𝜑 → (¬ 𝑊 ∈ V → 𝐹:𝑊⟶1-1On))
8	6, 7	mpd 16	. . . . 5 ⊢ (𝜑 → 𝐹:𝑊⟶1-1On)
9		f1f 6800	. . . . 5 ⊢ (𝐹:𝑊⟶1-1On → 𝐹:𝑊⟶On)
10		frn 6739	. . . . 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 3935	. . . . . . . . 9 ⊢ 𝑊 ⊆ 𝑊
14	4	onprcf1acwevdlem1 36045	. . . . . . . . 9 ⊢ ((𝑊 ⊆ 𝑊 ∧ 𝑣 ∈ On ∧ ∀𝑢 ∈ 𝑊 ¬ 𝑢 We (𝑅1‘𝑣)) → 𝑊 ∈ V)
15	13, 14	mp3an1 1478	. . . . . . . 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 3093	. . . . 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 6858	. . . . . . . . 9 ⊢ (𝐹:𝑊⟶1-1On → 𝐹:𝑊⟶1-1ontoran 𝐹)
22		f1ocnv 6859	. . . . . . . . 9 ⊢ (𝐹:𝑊⟶1-1ontoran 𝐹 → ◡𝐹:ran 𝐹⟶1-1onto𝑊)
23	8, 21, 22	3syl 19	. . . . . . . 8 ⊢ (𝜑 → ◡𝐹:ran 𝐹⟶1-1onto𝑊)
24		f1ofo 6854	. . . . . . . 8 ⊢ (◡𝐹:ran 𝐹⟶1-1onto𝑊 → ◡𝐹:ran 𝐹⟶onto𝑊)
25		forn 6821	. . . . . . . 8 ⊢ (◡𝐹:ran 𝐹⟶onto𝑊 → ran ◡𝐹 = 𝑊)
26	23, 24, 25	3syl 19	. . . . . . 7 ⊢ (𝜑 → ran ◡𝐹 = 𝑊)
27	26	rexeqdv 3324	. . . . . 6 ⊢ (𝜑 → (∃𝑢 ∈ ran ◡𝐹𝑢 We (𝑅1‘𝑣) ↔ ∃𝑢 ∈ 𝑊 𝑢 We (𝑅1‘𝑣)))
28		f1ofn 6847	. . . . . . 7 ⊢ (◡𝐹:ran 𝐹⟶1-1onto𝑊 → ◡𝐹 Fn ran 𝐹)
29		weeq1 5642	. . . . . . . 8 ⊢ (𝑢 = (◡𝐹‘𝑤) → (𝑢 We (𝑅1‘𝑣) ↔ (◡𝐹‘𝑤) We (𝑅1‘𝑣)))
30	29	rexrn 7111	. . . . . . 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 3983	. . 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 36046	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 4910   class class class wbr 5106  {copab 5170   We wwe 5607   × cxp 5653  ^◡ccnv 5654  ran crn 5656  Oncon0 6380  suc csuc 6382   Fn wfn 6553  ⟶wf 6554  ⟶1-1wf1 6555  ⟶ontowfo 6556  ⟶1-1ontowf1o 6557  ‘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:  onf1oracwevd  36052