Theorem oenass 43312

Description: Ordinal exponentiation is not associative. Remark 4.6 of [Schloeder] p. 14. (Contributed by RP, 30-Jan-2025.)

Assertion

Ref	Expression
oenass	⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐)

Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem oenass

Step	Hyp	Ref	Expression
1		oenassex 43311	. 2 ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)
2		2on 8401	. . 3 ⊢ 2o ∈ On
3		0elon 6362	. . . . 5 ⊢ ∅ ∈ On
4		oveq2 7357	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (2o ↑o 𝑐) = (2o ↑o ∅))
5	4	oveq2d 7365	. . . . . . . 8 ⊢ (𝑐 = ∅ → (2o ↑o (2o ↑o 𝑐)) = (2o ↑o (2o ↑o ∅)))
6		oveq2 7357	. . . . . . . 8 ⊢ (𝑐 = ∅ → ((2o ↑o 2o) ↑o 𝑐) = ((2o ↑o 2o) ↑o ∅))
7	5, 6	eqeq12d 2745	. . . . . . 7 ⊢ (𝑐 = ∅ → ((2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐) ↔ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)))
8	7	notbid 318	. . . . . 6 ⊢ (𝑐 = ∅ → (¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐) ↔ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)))
9	8	rspcev 3577	. . . . 5 ⊢ ((∅ ∈ On ∧ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)) → ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐))
10	3, 9	mpan 690	. . . 4 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐))
11		oveq1 7356	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ↑o 𝑐) = (2o ↑o 𝑐))
12	11	oveq2d 7365	. . . . . . . 8 ⊢ (𝑏 = 2o → (2o ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (2o ↑o 𝑐)))
13		oveq2 7357	. . . . . . . . 9 ⊢ (𝑏 = 2o → (2o ↑o 𝑏) = (2o ↑o 2o))
14	13	oveq1d 7364	. . . . . . . 8 ⊢ (𝑏 = 2o → ((2o ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 2o) ↑o 𝑐))
15	12, 14	eqeq12d 2745	. . . . . . 7 ⊢ (𝑏 = 2o → ((2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
16	15	notbid 318	. . . . . 6 ⊢ (𝑏 = 2o → (¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
17	16	rexbidv 3153	. . . . 5 ⊢ (𝑏 = 2o → (∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
18	17	rspcev 3577	. . . 4 ⊢ ((2o ∈ On ∧ ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐))
19	2, 10, 18	sylancr 587	. . 3 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐))
20		oveq1 7356	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (𝑏 ↑o 𝑐)))
21		oveq1 7356	. . . . . . . . 9 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o ↑o 𝑏))
22	21	oveq1d 7364	. . . . . . . 8 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 𝑏) ↑o 𝑐))
23	20, 22	eqeq12d 2745	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
24	23	notbid 318	. . . . . 6 ⊢ (𝑎 = 2o → (¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
25	24	rexbidv 3153	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
26	25	rexbidv 3153	. . . 4 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
27	26	rspcev 3577	. . 3 ⊢ ((2o ∈ On ∧ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐))
28	2, 19, 27	sylancr 587	. 2 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐))
29	1, 28	ax-mp 5	1 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ∅c0 4284  Oncon0 6307  (class class class)co 7349  2_oc2o 8382   ↑_o coe 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671  ax-reg 9484

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-oexp 8394

This theorem is referenced by: (None)