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Mirrors > Home > MPE Home > Th. List > alephsson | Structured version Visualization version GIF version |
Description: The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
alephsson | ⊢ ran ℵ ⊆ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinfcard 9671 | . . 3 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ 𝑥 ∈ ran ℵ) | |
2 | cardon 9525 | . . . . 5 ⊢ (card‘𝑥) ∈ On | |
3 | eleq1 2818 | . . . . 5 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝑥) ∈ On ↔ 𝑥 ∈ On)) | |
4 | 2, 3 | mpbii 236 | . . . 4 ⊢ ((card‘𝑥) = 𝑥 → 𝑥 ∈ On) |
5 | 4 | adantl 485 | . . 3 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) → 𝑥 ∈ On) |
6 | 1, 5 | sylbir 238 | . 2 ⊢ (𝑥 ∈ ran ℵ → 𝑥 ∈ On) |
7 | 6 | ssriv 3891 | 1 ⊢ ran ℵ ⊆ On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ran crn 5537 Oncon0 6191 ‘cfv 6358 ωcom 7622 cardccrd 9516 ℵcale 9517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-oi 9104 df-har 9151 df-card 9520 df-aleph 9521 |
This theorem is referenced by: unialeph 9680 alephsmo 9681 alephfplem3 9685 alephfp 9687 alephfp2 9688 |
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