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Mirrors > Home > MPE Home > Th. List > Mathboxes > alephiso3 | Structured version Visualization version GIF version |
Description: ℵ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.) |
Ref | Expression |
---|---|
alephiso3 | ⊢ ℵ Isom E , ≺ (On, (ran card ∖ ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephiso2 41837 | . 2 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) | |
2 | omelon 9583 | . . . . . 6 ⊢ ω ∈ On | |
3 | elrncard 41816 | . . . . . . 7 ⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥)) | |
4 | 3 | simplbi 499 | . . . . . 6 ⊢ (𝑥 ∈ ran card → 𝑥 ∈ On) |
5 | ontri1 6352 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω)) | |
6 | 2, 4, 5 | sylancr 588 | . . . . 5 ⊢ (𝑥 ∈ ran card → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω)) |
7 | 6 | rabbiia 3412 | . . . 4 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω} |
8 | dfdif2 3920 | . . . 4 ⊢ (ran card ∖ ω) = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω} | |
9 | 7, 8 | eqtr4i 2768 | . . 3 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω) |
10 | isoeq5 7267 | . . 3 ⊢ ({𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω) → (ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) ↔ ℵ Isom E , ≺ (On, (ran card ∖ ω)))) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) ↔ ℵ Isom E , ≺ (On, (ran card ∖ ω))) |
12 | 1, 11 | mpbi 229 | 1 ⊢ ℵ Isom E , ≺ (On, (ran card ∖ ω)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∀wral 3065 {crab 3408 ∖ cdif 3908 ⊆ wss 3911 class class class wbr 5106 E cep 5537 ran crn 5635 Oncon0 6318 Isom wiso 6498 ωcom 7803 ≈ cen 8881 ≺ csdm 8883 cardccrd 9872 ℵcale 9873 |
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-rep 5243 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-rmo 3354 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-se 5590 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-isom 6506 df-riota 7314 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-oi 9447 df-har 9494 df-card 9876 df-aleph 9877 |
This theorem is referenced by: (None) |
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