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Mirrors > Home > MPE Home > Th. List > alephf1ALT | Structured version Visualization version GIF version |
Description: Alternate proof of alephf1 9499. (Contributed by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
alephf1ALT | ⊢ ℵ:On–1-1→On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephfnon 9479 | . . 3 ⊢ ℵ Fn On | |
2 | alephon 9483 | . . . . 5 ⊢ (ℵ‘𝑥) ∈ On | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ On → (ℵ‘𝑥) ∈ On) |
4 | 3 | rgen 3145 | . . 3 ⊢ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On |
5 | ffnfv 6874 | . . 3 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ On)) | |
6 | 1, 4, 5 | mpbir2an 707 | . 2 ⊢ ℵ:On⟶On |
7 | alephsmo 9516 | . 2 ⊢ Smo ℵ | |
8 | smo11 7990 | . 2 ⊢ ((ℵ:On⟶On ∧ Smo ℵ) → ℵ:On–1-1→On) | |
9 | 6, 7, 8 | mp2an 688 | 1 ⊢ ℵ:On–1-1→On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∀wral 3135 Oncon0 6184 Fn wfn 6343 ⟶wf 6344 –1-1→wf1 6345 ‘cfv 6348 Smo wsmo 7971 ℵcale 9353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-om 7570 df-wrecs 7936 df-smo 7972 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-oi 8962 df-har 9010 df-card 9356 df-aleph 9357 |
This theorem is referenced by: (None) |
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