Theorem wevsev 35953

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 3932	. . . . 5 ⊢ {𝑧 ∈ V ∣ (rank‘𝑧) = (rank‘𝑤)} ⊆ V
2		wess 5636	. . . . 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 3158	. . 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 35785	. . 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 35786	. 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 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102  {copab 5166   Se wse 5601   We wwe 5602  ‘cfv 6538  rankcrnk 9768

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-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-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-r1 9769  df-rank 9770

This theorem is used by: (None)