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Mirrors > Home > MPE Home > Th. List > Mathboxes > sucomisnotcard | Structured version Visualization version GIF version |
Description: ω +o 1o is not a cardinal number. (Contributed by RP, 1-Oct-2023.) |
Ref | Expression |
---|---|
sucomisnotcard | ⊢ ¬ (ω +o 1o) ∈ ran card |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omelon 9583 | . . . . . 6 ⊢ ω ∈ On | |
2 | sucidg 6399 | . . . . . 6 ⊢ (ω ∈ On → ω ∈ suc ω) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ω ∈ suc ω |
4 | omensuc 9593 | . . . . 5 ⊢ ω ≈ suc ω | |
5 | breq1 5109 | . . . . . 6 ⊢ (𝑥 = ω → (𝑥 ≈ suc ω ↔ ω ≈ suc ω)) | |
6 | 5 | rspcev 3582 | . . . . 5 ⊢ ((ω ∈ suc ω ∧ ω ≈ suc ω) → ∃𝑥 ∈ suc ω𝑥 ≈ suc ω) |
7 | 3, 4, 6 | mp2an 691 | . . . 4 ⊢ ∃𝑥 ∈ suc ω𝑥 ≈ suc ω |
8 | dfrex2 3077 | . . . 4 ⊢ (∃𝑥 ∈ suc ω𝑥 ≈ suc ω ↔ ¬ ∀𝑥 ∈ suc ω ¬ 𝑥 ≈ suc ω) | |
9 | 7, 8 | mpbi 229 | . . 3 ⊢ ¬ ∀𝑥 ∈ suc ω ¬ 𝑥 ≈ suc ω |
10 | 9 | intnan 488 | . 2 ⊢ ¬ (suc ω ∈ On ∧ ∀𝑥 ∈ suc ω ¬ 𝑥 ≈ suc ω) |
11 | oa1suc 8478 | . . . . 5 ⊢ (ω ∈ On → (ω +o 1o) = suc ω) | |
12 | 1, 11 | ax-mp 5 | . . . 4 ⊢ (ω +o 1o) = suc ω |
13 | 12 | eleq1i 2829 | . . 3 ⊢ ((ω +o 1o) ∈ ran card ↔ suc ω ∈ ran card) |
14 | elrncard 41816 | . . 3 ⊢ (suc ω ∈ ran card ↔ (suc ω ∈ On ∧ ∀𝑥 ∈ suc ω ¬ 𝑥 ≈ suc ω)) | |
15 | 13, 14 | sylbb 218 | . 2 ⊢ ((ω +o 1o) ∈ ran card → (suc ω ∈ On ∧ ∀𝑥 ∈ suc ω ¬ 𝑥 ≈ suc ω)) |
16 | 10, 15 | mto 196 | 1 ⊢ ¬ (ω +o 1o) ∈ ran card |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∃wrex 3074 class class class wbr 5106 ran crn 5635 Oncon0 6318 suc csuc 6320 (class class class)co 7358 ωcom 7803 1oc1o 8406 +o coa 8410 ≈ cen 8881 cardccrd 9872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-card 9876 |
This theorem is referenced by: (None) |
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