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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 43053 | . 2 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) | |
2 | omelon 9669 | . . . . . 6 ⊢ ω ∈ On | |
3 | elrncard 43032 | . . . . . . 7 ⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥)) | |
4 | 3 | simplbi 496 | . . . . . 6 ⊢ (𝑥 ∈ ran card → 𝑥 ∈ On) |
5 | ontri1 6398 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω)) | |
6 | 2, 4, 5 | sylancr 585 | . . . . 5 ⊢ (𝑥 ∈ ran card → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω)) |
7 | 6 | rabbiia 3423 | . . . 4 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω} |
8 | dfdif2 3948 | . . . 4 ⊢ (ran card ∖ ω) = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω} | |
9 | 7, 8 | eqtr4i 2756 | . . 3 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω) |
10 | isoeq5 7325 | . . 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 1533 ∈ wcel 2098 ∀wral 3051 {crab 3419 ∖ cdif 3936 ⊆ wss 3939 class class class wbr 5143 E cep 5575 ran crn 5673 Oncon0 6364 Isom wiso 6544 ωcom 7868 ≈ cen 8959 ≺ csdm 8961 cardccrd 9958 ℵcale 9959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-oi 9533 df-har 9580 df-card 9962 df-aleph 9963 |
This theorem is referenced by: (None) |
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