New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > tlenc1c | GIF version |
Description: A T-raising is less than or equal to the cardinality of cardinal one. (Contributed by SF, 16-Mar-2015.) |
Ref | Expression |
---|---|
tlenc1c | ⊢ (M ∈ NC → Tc M ≤c Nc 1c) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elncs 6119 | . 2 ⊢ (M ∈ NC ↔ ∃x M = Nc x) | |
2 | tceq 6158 | . . . . 5 ⊢ (M = Nc x → Tc M = Tc Nc x) | |
3 | vex 2862 | . . . . . 6 ⊢ x ∈ V | |
4 | 3 | tcnc 6225 | . . . . 5 ⊢ Tc Nc x = Nc ℘1x |
5 | 2, 4 | syl6eq 2401 | . . . 4 ⊢ (M = Nc x → Tc M = Nc ℘1x) |
6 | 3 | pw1ex 4303 | . . . . . . 7 ⊢ ℘1x ∈ V |
7 | 6 | ncid 6123 | . . . . . 6 ⊢ ℘1x ∈ Nc ℘1x |
8 | 1cex 4142 | . . . . . . 7 ⊢ 1c ∈ V | |
9 | 8 | ncid 6123 | . . . . . 6 ⊢ 1c ∈ Nc 1c |
10 | pw1ss1c 4158 | . . . . . 6 ⊢ ℘1x ⊆ 1c | |
11 | sseq1 3292 | . . . . . . 7 ⊢ (y = ℘1x → (y ⊆ z ↔ ℘1x ⊆ z)) | |
12 | sseq2 3293 | . . . . . . 7 ⊢ (z = 1c → (℘1x ⊆ z ↔ ℘1x ⊆ 1c)) | |
13 | 11, 12 | rspc2ev 2963 | . . . . . 6 ⊢ ((℘1x ∈ Nc ℘1x ∧ 1c ∈ Nc 1c ∧ ℘1x ⊆ 1c) → ∃y ∈ Nc ℘1x∃z ∈ Nc 1cy ⊆ z) |
14 | 7, 9, 10, 13 | mp3an 1277 | . . . . 5 ⊢ ∃y ∈ Nc ℘1x∃z ∈ Nc 1cy ⊆ z |
15 | ncex 6117 | . . . . . 6 ⊢ Nc ℘1x ∈ V | |
16 | ncex 6117 | . . . . . 6 ⊢ Nc 1c ∈ V | |
17 | 15, 16 | brlec 6113 | . . . . 5 ⊢ ( Nc ℘1x ≤c Nc 1c ↔ ∃y ∈ Nc ℘1x∃z ∈ Nc 1cy ⊆ z) |
18 | 14, 17 | mpbir 200 | . . . 4 ⊢ Nc ℘1x ≤c Nc 1c |
19 | 5, 18 | syl6eqbr 4676 | . . 3 ⊢ (M = Nc x → Tc M ≤c Nc 1c) |
20 | 19 | exlimiv 1634 | . 2 ⊢ (∃x M = Nc x → Tc M ≤c Nc 1c) |
21 | 1, 20 | sylbi 187 | 1 ⊢ (M ∈ NC → Tc M ≤c Nc 1c) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ⊆ wss 3257 1cc1c 4134 ℘1cpw1 4135 class class class wbr 4639 NC cncs 6088 ≤c clec 6089 Nc cnc 6091 Tc ctc 6093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-2nd 4797 df-txp 5736 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-en 6029 df-ncs 6098 df-lec 6099 df-nc 6101 df-tc 6103 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |