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Mirrors > Home > MPE Home > Th. List > Mathboxes > oenord1ex | Structured version Visualization version GIF version |
Description: When ordinals two and three are both raised to the power of omega, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.) |
Ref | Expression |
---|---|
oenord1ex | ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2oex 8516 | . . . . 5 ⊢ 2o ∈ V | |
2 | 1 | tpid3 4778 | . . . 4 ⊢ 2o ∈ {∅, 1o, 2o} |
3 | df3o2 43303 | . . . 4 ⊢ 3o = {∅, 1o, 2o} | |
4 | 2, 3 | eleqtrri 2838 | . . 3 ⊢ 2o ∈ 3o |
5 | ordom 7897 | . . . 4 ⊢ Ord ω | |
6 | ordirr 6404 | . . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω) | |
7 | 2onn 8679 | . . . . . . 7 ⊢ 2o ∈ ω | |
8 | 1oex 8515 | . . . . . . . . 9 ⊢ 1o ∈ V | |
9 | 8 | prid2 4768 | . . . . . . . 8 ⊢ 1o ∈ {∅, 1o} |
10 | df2o3 8513 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
11 | 9, 10 | eleqtrri 2838 | . . . . . . 7 ⊢ 1o ∈ 2o |
12 | nnoeomeqom 43302 | . . . . . . 7 ⊢ ((2o ∈ ω ∧ 1o ∈ 2o) → (2o ↑o ω) = ω) | |
13 | 7, 11, 12 | mp2an 692 | . . . . . 6 ⊢ (2o ↑o ω) = ω |
14 | 3onn 8681 | . . . . . . 7 ⊢ 3o ∈ ω | |
15 | 8 | tpid2 4775 | . . . . . . . 8 ⊢ 1o ∈ {∅, 1o, 2o} |
16 | 15, 3 | eleqtrri 2838 | . . . . . . 7 ⊢ 1o ∈ 3o |
17 | nnoeomeqom 43302 | . . . . . . 7 ⊢ ((3o ∈ ω ∧ 1o ∈ 3o) → (3o ↑o ω) = ω) | |
18 | 14, 16, 17 | mp2an 692 | . . . . . 6 ⊢ (3o ↑o ω) = ω |
19 | 13, 18 | eleq12i 2832 | . . . . 5 ⊢ ((2o ↑o ω) ∈ (3o ↑o ω) ↔ ω ∈ ω) |
20 | 6, 19 | sylnibr 329 | . . . 4 ⊢ (Ord ω → ¬ (2o ↑o ω) ∈ (3o ↑o ω)) |
21 | 5, 20 | ax-mp 5 | . . 3 ⊢ ¬ (2o ↑o ω) ∈ (3o ↑o ω) |
22 | 4, 21 | 2th 264 | . 2 ⊢ (2o ∈ 3o ↔ ¬ (2o ↑o ω) ∈ (3o ↑o ω)) |
23 | xor3 382 | . 2 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) ↔ (2o ∈ 3o ↔ ¬ (2o ↑o ω) ∈ (3o ↑o ω))) | |
24 | 22, 23 | mpbir 231 | 1 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∅c0 4339 {cpr 4633 {ctp 4635 Ord word 6385 (class class class)co 7431 ωcom 7887 1oc1o 8498 2oc2o 8499 3oc3o 8500 ↑o coe 8504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-3o 8507 df-oadd 8509 df-omul 8510 df-oexp 8511 |
This theorem is referenced by: oenord1 43306 |
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