Theorem onprcf1of1orbd 35966

Description: If 𝐹 is a bijection from the ordinals to the universe and both 𝐴 and 𝐵 are sets, then either 𝐴 can be mapped one-to-one into 𝐵 or 𝐵 can be mapped one-to-one into 𝐴. This is the ZFC version of (5 → 10) in https://tinyurl.com/hamkins-gblac in the case where neither class is a proper class. See onprcf1of1orad 35965 for the case where one of the classes is a proper class, and see onprcf1of1ord 35967 for a theorem that covers both cases. Note that in NBG set theory the first hypothesis would be replaced by something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On–1-1-onto→𝑋)), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 17-Sep-2026.)

Hypotheses

Ref	Expression
onprcf1of1orbd.1	⊢ (𝜑 → 𝐹:On–1-1-onto→V)
onprcf1of1orbd.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
onprcf1of1orbd.3	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
onprcf1of1orbd	⊢ (𝜑 → (∃𝑓 𝑓:𝐴–1-1→𝐵 ∨ ∃ℎ ℎ:𝐵–1-1→𝐴))

Distinct variable groups:   𝐴,𝑓   𝐴,ℎ   𝐵,𝑓   𝐵,ℎ

Allowed substitution hints:   𝜑(𝑓, ℎ)   𝐹(𝑓, ℎ)   𝑉(𝑓, ℎ)   𝑊(𝑓, ℎ)

Proof of Theorem onprcf1of1orbd

Dummy variables 𝑥 𝑧 𝑖 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onprcf1of1orbd.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:On–1-1-onto→V)
2		f1ocnv 6837	. . . . . . . 8 ⊢ (𝐹:On–1-1-onto→V → ◡𝐹:V–1-1-onto→On)
3		f1ofun 6826	. . . . . . . 8 ⊢ (◡𝐹:V–1-1-onto→On → Fun ◡𝐹)
4		resfunexg 7221	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ 𝑧 ∈ V) → (◡𝐹 ↾ 𝑧) ∈ V)
5	4	elvd 3456	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (◡𝐹 ↾ 𝑧) ∈ V)
6	2, 3, 5	3syl 19	. . . . . . 7 ⊢ (𝐹:On–1-1-onto→V → (◡𝐹 ↾ 𝑧) ∈ V)
7	1, 6	syl 18	. . . . . 6 ⊢ (𝜑 → (◡𝐹 ↾ 𝑧) ∈ V)
8		f1of1 6823	. . . . . . . 8 ⊢ (◡𝐹:V–1-1-onto→On → ◡𝐹:V–1-1→On)
9		ssv 3932	. . . . . . . 8 ⊢ 𝑧 ⊆ V
10		f1ssres 6787	. . . . . . . 8 ⊢ ((◡𝐹:V–1-1→On ∧ 𝑧 ⊆ V) → (◡𝐹 ↾ 𝑧):𝑧–1-1→On)
11	8, 9, 10	sylancl 598	. . . . . . 7 ⊢ (◡𝐹:V–1-1-onto→On → (◡𝐹 ↾ 𝑧):𝑧–1-1→On)
12	1, 2, 11	3syl 19	. . . . . 6 ⊢ (𝜑 → (◡𝐹 ↾ 𝑧):𝑧–1-1→On)
13		f1eq1 6773	. . . . . 6 ⊢ (𝑖 = (◡𝐹 ↾ 𝑧) → (𝑖:𝑧–1-1→On ↔ (◡𝐹 ↾ 𝑧):𝑧–1-1→On))
14	7, 12, 13	spcedv 3552	. . . . 5 ⊢ (𝜑 → ∃𝑖 𝑖:𝑧–1-1→On)
15	14	alrimiv 1959	. . . 4 ⊢ (𝜑 → ∀𝑧∃𝑖 𝑖:𝑧–1-1→On)
16		dfac15 35802	. . . 4 ⊢ (CHOICE ↔ ∀𝑧∃𝑖 𝑖:𝑧–1-1→On)
17	15, 16	sylibr 237	. . 3 ⊢ (𝜑 → CHOICE)
18		dfac16 35803	. . . 4 ⊢ (CHOICE ↔ ∀𝑥∀𝑦(𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥))
19		onprcf1of1orbd.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
20		onprcf1of1orbd.3	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
21		breq2 5106	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑥 ≼ 𝑦 ↔ 𝑥 ≼ 𝐵))
22		breq1 5105	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑦 ≼ 𝑥 ↔ 𝐵 ≼ 𝑥))
23	21, 22	orbi12d 932	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → ((𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥) ↔ (𝑥 ≼ 𝐵 ∨ 𝐵 ≼ 𝑥)))
24	23	spcgv 3550	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (∀𝑦(𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥) → (𝑥 ≼ 𝐵 ∨ 𝐵 ≼ 𝑥)))
25	20, 24	syl 18	. . . . . 6 ⊢ (𝜑 → (∀𝑦(𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥) → (𝑥 ≼ 𝐵 ∨ 𝐵 ≼ 𝑥)))
26	25	alimdv 1948	. . . . 5 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥) → ∀𝑥(𝑥 ≼ 𝐵 ∨ 𝐵 ≼ 𝑥)))
27		breq1 5105	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ≼ 𝐵 ↔ 𝐴 ≼ 𝐵))
28		breq2 5106	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 ≼ 𝑥 ↔ 𝐵 ≼ 𝐴))
29	27, 28	orbi12d 932	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ≼ 𝐵 ∨ 𝐵 ≼ 𝑥) ↔ (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)))
30	29	spcgv 3550	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ≼ 𝐵 ∨ 𝐵 ≼ 𝑥) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)))
31	19, 26, 30	sylsyld 62	. . . 4 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)))
32	18, 31	biimtrid 245	. . 3 ⊢ (𝜑 → (CHOICE → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)))
33	17, 32	mpd 16	. 2 ⊢ (𝜑 → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴))
34		brdomg 8986	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
35	20, 34	syl 18	. . 3 ⊢ (𝜑 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
36		brdomg 8986	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ≼ 𝐴 ↔ ∃ℎ ℎ:𝐵–1-1→𝐴))
37	19, 36	syl 18	. . 3 ⊢ (𝜑 → (𝐵 ≼ 𝐴 ↔ ∃ℎ ℎ:𝐵–1-1→𝐴))
38	35, 37	orbi12d 932	. 2 ⊢ (𝜑 → ((𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴–1-1→𝐵 ∨ ∃ℎ ℎ:𝐵–1-1→𝐴)))
39	33, 38	mpbid 235	1 ⊢ (𝜑 → (∃𝑓 𝑓:𝐴–1-1→𝐵 ∨ ∃ℎ ℎ:𝐵–1-1→𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102  ^◡ccnv 5649   ↾ cres 5652  Oncon0 6362  Fun wfun 6532  –1-1→wf1 6535  –1-1-onto→wf1o 6537   ≼ cdom 8972  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

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-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-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-er 8718  df-en 8975  df-dom 8976  df-sdom 8977  df-oi 9505  df-har 9552  df-card 10021  df-ac 10196

This theorem is used by:  onprcf1of1ord  35967