Theorem oenass 43773

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 43772	. 2 ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)
2		2on 8415	. . 3 ⊢ 2o ∈ On
3		0elon 6376	. . . . 5 ⊢ ∅ ∈ On
4		oveq2 7372	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (2o ↑o 𝑐) = (2o ↑o ∅))
5	4	oveq2d 7380	. . . . . . . 8 ⊢ (𝑐 = ∅ → (2o ↑o (2o ↑o 𝑐)) = (2o ↑o (2o ↑o ∅)))
6		oveq2 7372	. . . . . . . 8 ⊢ (𝑐 = ∅ → ((2o ↑o 2o) ↑o 𝑐) = ((2o ↑o 2o) ↑o ∅))
7	5, 6	eqeq12d 2753	. . . . . . 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 3565	. . . . 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 691	. . . 4 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐))
11		oveq1 7371	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ↑o 𝑐) = (2o ↑o 𝑐))
12	11	oveq2d 7380	. . . . . . . 8 ⊢ (𝑏 = 2o → (2o ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (2o ↑o 𝑐)))
13		oveq2 7372	. . . . . . . . 9 ⊢ (𝑏 = 2o → (2o ↑o 𝑏) = (2o ↑o 2o))
14	13	oveq1d 7379	. . . . . . . 8 ⊢ (𝑏 = 2o → ((2o ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 2o) ↑o 𝑐))
15	12, 14	eqeq12d 2753	. . . . . . 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 3162	. . . . 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 3565	. . . 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 588	. . 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 7371	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (𝑏 ↑o 𝑐)))
21		oveq1 7371	. . . . . . . . 9 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o ↑o 𝑏))
22	21	oveq1d 7379	. . . . . . . 8 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 𝑏) ↑o 𝑐))
23	20, 22	eqeq12d 2753	. . . . . . 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 3162	. . . . 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 3162	. . . 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 3565	. . 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 588	. 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 1542   ∈ wcel 2114  ∃wrex 3062  ∅c0 4274  Oncon0 6321  (class class class)co 7364  2_oc2o 8396   ↑_o coe 8401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5374  ax-un 7686  ax-reg 9504

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-omul 8407  df-oexp 8408

This theorem is referenced by: (None)