Theorem j0iso 35936

Description: 𝐽₀ is the 𝑅₀ order isomorphism of On × On onto On. Definition 7.59 of [TakeutiZaring] p. 55. This isomorphism is unique by weisoeq 7365 and r0weseon 35932. (Contributed by BTernaryTau, 19-Sep-2026.)

Assertion

Ref	Expression
j0iso	⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)

Proof of Theorem j0iso

Step	Hyp	Ref	Expression
1		r0weseon 35932	. . 3 ⊢ (𝑅0 We (On × On) ∧ 𝑅0 Se (On × On))
2		onprc 7792	. . . . 5 ⊢ ¬ On ∈ V
3		xpexr 7930	. . . . . 6 ⊢ ((On × On) ∈ V → (On ∈ V ∨ On ∈ V))
4		pm1.2 917	. . . . . 6 ⊢ ((On ∈ V ∨ On ∈ V) → On ∈ V)
5	3, 4	syl 18	. . . . 5 ⊢ ((On × On) ∈ V → On ∈ V)
6	2, 5	mto 200	. . . 4 ⊢ ¬ (On × On) ∈ V
7		eqid 2760	. . . . 5 ⊢ OrdIso(𝑅0, (On × On)) = OrdIso(𝑅0, (On × On))
8	7	ordtypeon 35763	. . . 4 ⊢ ((𝑅0 We (On × On) ∧ 𝑅0 Se (On × On) ∧ ¬ (On × On) ∈ V) → OrdIso(𝑅0, (On × On)) Isom E , 𝑅0 (On, (On × On)))
9	6, 8	mp3an3 1479	. . 3 ⊢ ((𝑅0 We (On × On) ∧ 𝑅0 Se (On × On)) → OrdIso(𝑅0, (On × On)) Isom E , 𝑅0 (On, (On × On)))
10		isocnv 7338	. . 3 ⊢ (OrdIso(𝑅0, (On × On)) Isom E , 𝑅0 (On, (On × On)) → ◡OrdIso(𝑅0, (On × On)) Isom 𝑅0, E ((On × On), On))
11	1, 9, 10	mp2b 10	. 2 ⊢ ◡OrdIso(𝑅0, (On × On)) Isom 𝑅0, E ((On × On), On)
12		df-j0 35923	. . 3 ⊢ 𝐽0 = ◡OrdIso(𝑅0, (On × On))
13		isoeq1 7325	. . 3 ⊢ (𝐽0 = ◡OrdIso(𝑅0, (On × On)) → (𝐽0 Isom 𝑅0, E ((On × On), On) ↔ ◡OrdIso(𝑅0, (On × On)) Isom 𝑅0, E ((On × On), On)))
14	12, 13	ax-mp 5	. 2 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) ↔ ◡OrdIso(𝑅0, (On × On)) Isom 𝑅0, E ((On × On), On))
15	11, 14	mpbir 234	1 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  Vcvv 3450   E cep 5549   Se wse 5601   We wwe 5602   × cxp 5648  ^◡ccnv 5649  Oncon0 6362   Isom wiso 6539  OrdIsocoi 9504  𝑅₀cr0 35913  𝐽₀ccj0 35914

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-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-oi 9505  df-lexo 35921  df-r0 35922  df-j0 35923

This theorem is used by:  j00smo  35937  maxssj0  35938  j0ltalephlem  35939  j0ltaleph  35940