Theorem j0iso 36004

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 7388 and r0weseon 36000. (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 36000	. . 3 ⊢ (𝑅0 We (On × On) ∧ 𝑅0 Se (On × On))
2		onprc 7816	. . . . 5 ⊢ ¬ On ∈ V
3		xpexr 7954	. . . . . 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 2763	. . . . 5 ⊢ OrdIso(𝑅0, (On × On)) = OrdIso(𝑅0, (On × On))
8	7	ordtypeon 35802	. . . 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 1480	. . 3 ⊢ ((𝑅0 We (On × On) ∧ 𝑅0 Se (On × On)) → OrdIso(𝑅0, (On × On)) Isom E , 𝑅0 (On, (On × On)))
10		isocnv 7361	. . 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 35991	. . 3 ⊢ 𝐽0 = ◡OrdIso(𝑅0, (On × On))
13		isoeq1 7348	. . 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 1570   ∈ wcel 2147  Vcvv 3454   E cep 5554   Se wse 5606   We wwe 5607   × cxp 5653  ^◡ccnv 5654  Oncon0 6380   Isom wiso 6559  OrdIsocoi 9530  𝑅₀cr0 35981  𝐽₀ccj0 35982

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-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-oi 9531  df-lexo 35989  df-r0 35990  df-j0 35991

This theorem is used by:  j00smo  36005  maxssj0  36006  j0ltalephlem  36007  j0ltaleph  36008  jf1  36010  jf1o  36011  j0le  36019  k3n0j0lt  36020  jltalephlem  36025  jltaleph  36026