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Theorem oenassex 41903
Description: Ordinal two raised to two to the zeroth power is not the same as two squared then raised to the zeroth power. (Contributed by RP, 30-Jan-2025.)
Assertion
Ref Expression
oenassex ¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅)

Proof of Theorem oenassex
StepHypRef Expression
1 1oex 8460 . . . 4 1o ∈ V
21prid2 4761 . . 3 1o ∈ {∅, 1o}
3 df2o3 8458 . . 3 2o = {∅, 1o}
42, 3eleqtrri 2832 . 2 1o ∈ 2o
5 elneq 9577 . . 3 (1o ∈ 2o → 1o ≠ 2o)
6 df-ne 2941 . . . 4 (2o ≠ 1o ↔ ¬ 2o = 1o)
7 necom 2994 . . . 4 (1o ≠ 2o ↔ 2o ≠ 1o)
8 2on 8464 . . . . . . . . 9 2o ∈ On
9 oe0 8506 . . . . . . . . 9 (2o ∈ On → (2oo ∅) = 1o)
108, 9ax-mp 5 . . . . . . . 8 (2oo ∅) = 1o
1110oveq2i 7405 . . . . . . 7 (2oo (2oo ∅)) = (2oo 1o)
12 oe1 8529 . . . . . . . 8 (2o ∈ On → (2oo 1o) = 2o)
138, 12ax-mp 5 . . . . . . 7 (2oo 1o) = 2o
1411, 13eqtri 2760 . . . . . 6 (2oo (2oo ∅)) = 2o
158, 8pm3.2i 471 . . . . . . 7 (2o ∈ On ∧ 2o ∈ On)
16 oecl 8521 . . . . . . 7 ((2o ∈ On ∧ 2o ∈ On) → (2oo 2o) ∈ On)
17 oe0 8506 . . . . . . 7 ((2oo 2o) ∈ On → ((2oo 2o) ↑o ∅) = 1o)
1815, 16, 17mp2b 10 . . . . . 6 ((2oo 2o) ↑o ∅) = 1o
1914, 18eqeq12i 2750 . . . . 5 ((2oo (2oo ∅)) = ((2oo 2o) ↑o ∅) ↔ 2o = 1o)
2019notbii 319 . . . 4 (¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅) ↔ ¬ 2o = 1o)
216, 7, 203bitr4i 302 . . 3 (1o ≠ 2o ↔ ¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅))
225, 21sylib 217 . 2 (1o ∈ 2o → ¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅))
234, 22ax-mp 5 1 ¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1541  wcel 2106  wne 2940  c0 4319  {cpr 4625  Oncon0 6354  (class class class)co 7394  1oc1o 8443  2oc2o 8444  o coe 8449
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709  ax-reg 9571
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-oadd 8454  df-omul 8455  df-oexp 8456
This theorem is referenced by:  oenass  41904
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