Theorem wevsev 36041

Description: If 𝑅 is a well-ordering of the universe, then 𝑆 is a set-like well-ordering of the universe. This is the ZFC version of (3 → 4) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 18-Sep-2026.)

Hypothesis

Ref	Expression
wevsev.1	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}

Assertion

Ref	Expression
wevsev	⊢ (𝑅 We V → (𝑆 We V ∧ 𝑆 Se V))

Distinct variable group:   𝑥,𝑦,𝑅

Allowed substitution hints:   𝑆(𝑥, 𝑦)

Proof of Theorem wevsev

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3937	. . . . 5 ⊢ {𝑧 ∈ V ∣ (rank‘𝑧) = (rank‘𝑤)} ⊆ V
2		wess 5641	. . . . 5 ⊢ ({𝑧 ∈ V ∣ (rank‘𝑧) = (rank‘𝑤)} ⊆ V → (𝑅 We V → 𝑅 We {𝑧 ∈ V ∣ (rank‘𝑧) = (rank‘𝑤)}))
3	1, 2	ax-mp 5	. . . 4 ⊢ (𝑅 We V → 𝑅 We {𝑧 ∈ V ∣ (rank‘𝑧) = (rank‘𝑤)})
4	3	ralrimivw 3162	. . 3 ⊢ (𝑅 We V → ∀𝑤 ∈ V 𝑅 We {𝑧 ∈ V ∣ (rank‘𝑧) = (rank‘𝑤)})
5		wevsev.1	. . . 4 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
6	5	werankwe2 35822	. . 3 ⊢ (∀𝑤 ∈ V 𝑅 We {𝑧 ∈ V ∣ (rank‘𝑧) = (rank‘𝑤)} → 𝑆 We V)
7	4, 6	syl 18	. 2 ⊢ (𝑅 We V → 𝑆 We V)
8	5	rankse 35823	. 2 ⊢ 𝑆 Se V
9	7, 8	jctir 530	1 ⊢ (𝑅 We V → (𝑆 We V ∧ 𝑆 Se V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106  {copab 5170   Se wse 5606   We wwe 5607  ‘cfv 6558  rankcrnk 9794

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-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-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-r1 9795  df-rank 9796

This theorem is used by: (None)