Theorem onprcf1of1orbd 36054

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 36053 for the case where one of the classes is a proper class, and see onprcf1of1ord 36055 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⟶1-1onto𝑋)), 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⟶1-1ontoV)
onprcf1of1orbd.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
onprcf1of1orbd.3	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

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

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⟶1-1ontoV)
2		f1ocnv 6859	. . . . . . . 8 ⊢ (𝐹:On⟶1-1ontoV → ◡𝐹:V⟶1-1ontoOn)
3		f1ofun 6848	. . . . . . . 8 ⊢ (◡𝐹:V⟶1-1ontoOn → Fun ◡𝐹)
4		resfunexg 7244	. . . . . . . . 9 ⊢ ((Fun ◡𝐹 ∧ 𝑧 ∈ V) → (◡𝐹 ↾ 𝑧) ∈ V)
5	4	elvd 3460	. . . . . . . 8 ⊢ (Fun ◡𝐹 → (◡𝐹 ↾ 𝑧) ∈ V)
6	2, 3, 5	3syl 19	. . . . . . 7 ⊢ (𝐹:On⟶1-1ontoV → (◡𝐹 ↾ 𝑧) ∈ V)
7	1, 6	syl 18	. . . . . 6 ⊢ (𝜑 → (◡𝐹 ↾ 𝑧) ∈ V)
8		f1of1 6845	. . . . . . . 8 ⊢ (◡𝐹:V⟶1-1ontoOn → ◡𝐹:V⟶1-1On)
9		ssv 3937	. . . . . . . 8 ⊢ 𝑧 ⊆ V
10		f1ssres 6809	. . . . . . . 8 ⊢ ((◡𝐹:V⟶1-1On ∧ 𝑧 ⊆ V) → (◡𝐹 ↾ 𝑧):𝑧⟶1-1On)
11	8, 9, 10	sylancl 598	. . . . . . 7 ⊢ (◡𝐹:V⟶1-1ontoOn → (◡𝐹 ↾ 𝑧):𝑧⟶1-1On)
12	1, 2, 11	3syl 19	. . . . . 6 ⊢ (𝜑 → (◡𝐹 ↾ 𝑧):𝑧⟶1-1On)
13		f1eq1 6795	. . . . . 6 ⊢ (𝑖 = (◡𝐹 ↾ 𝑧) → (𝑖:𝑧⟶1-1On ↔ (◡𝐹 ↾ 𝑧):𝑧⟶1-1On))
14	7, 12, 13	spcedv 3556	. . . . 5 ⊢ (𝜑 → ∃𝑖 𝑖:𝑧⟶1-1On)
15	14	alrimiv 1962	. . . 4 ⊢ (𝜑 → ∀𝑧∃𝑖 𝑖:𝑧⟶1-1On)
16		dfac15 35845	. . . 4 ⊢ (CHOICE ↔ ∀𝑧∃𝑖 𝑖:𝑧⟶1-1On)
17	15, 16	sylibr 237	. . 3 ⊢ (𝜑 → CHOICE)
18		dfac16 35846	. . . 4 ⊢ (CHOICE ↔ ∀𝑥∀𝑦(𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥))
19		onprcf1of1orbd.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
20		onprcf1of1orbd.3	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
21		breq2 5110	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑥 ≼ 𝑦 ↔ 𝑥 ≼ 𝐵))
22		breq1 5109	. . . . . . . . 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 3554	. . . . . . 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 1951	. . . . 5 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥) → ∀𝑥(𝑥 ≼ 𝐵 ∨ 𝐵 ≼ 𝑥)))
27		breq1 5109	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ≼ 𝐵 ↔ 𝐴 ≼ 𝐵))
28		breq2 5110	. . . . . . 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 3554	. . . . 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 9013	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1𝐵))
35	20, 34	syl 18	. . 3 ⊢ (𝜑 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1𝐵))
36		brdomg 9013	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ≼ 𝐴 ↔ ∃ℎ ℎ:𝐵⟶1-1𝐴))
37	19, 36	syl 18	. . 3 ⊢ (𝜑 → (𝐵 ≼ 𝐴 ↔ ∃ℎ ℎ:𝐵⟶1-1𝐴))
38	35, 37	orbi12d 932	. 2 ⊢ (𝜑 → ((𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴⟶1-1𝐵 ∨ ∃ℎ ℎ:𝐵⟶1-1𝐴)))
39	33, 38	mpbid 235	1 ⊢ (𝜑 → (∃𝑓 𝑓:𝐴⟶1-1𝐵 ∨ ∃ℎ ℎ:𝐵⟶1-1𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106  ^◡ccnv 5654   ↾ cres 5657  Oncon0 6380  Fun wfun 6552  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557   ≼ cdom 8999  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

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-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-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-oi 9531  df-har 9578  df-card 10047  df-ac 10222

This theorem is used by:  onprcf1of1ord  36055