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Mirrors > Home > NFE Home > Th. List > addlec | GIF version |
Description: For non-empty sets, cardinal sum always increases cardinal less than or equal. Theorem XI.2.19 of [Rosser] p. 376. (Contributed by SF, 11-Mar-2015.) |
Ref | Expression |
---|---|
addlec | ⊢ ((M ∈ V ∧ N ∈ W ∧ (M +c N) ≠ ∅) → M ≤c (M +c N)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eladdc 4398 | . . . . . . 7 ⊢ (z ∈ (M +c N) ↔ ∃x ∈ M ∃y ∈ N ((x ∩ y) = ∅ ∧ z = (x ∪ y))) | |
2 | ssun1 3426 | . . . . . . . . . . 11 ⊢ x ⊆ (x ∪ y) | |
3 | sseq2 3293 | . . . . . . . . . . 11 ⊢ (z = (x ∪ y) → (x ⊆ z ↔ x ⊆ (x ∪ y))) | |
4 | 2, 3 | mpbiri 224 | . . . . . . . . . 10 ⊢ (z = (x ∪ y) → x ⊆ z) |
5 | 4 | adantl 452 | . . . . . . . . 9 ⊢ (((x ∩ y) = ∅ ∧ z = (x ∪ y)) → x ⊆ z) |
6 | 5 | rexlimivw 2734 | . . . . . . . 8 ⊢ (∃y ∈ N ((x ∩ y) = ∅ ∧ z = (x ∪ y)) → x ⊆ z) |
7 | 6 | reximi 2721 | . . . . . . 7 ⊢ (∃x ∈ M ∃y ∈ N ((x ∩ y) = ∅ ∧ z = (x ∪ y)) → ∃x ∈ M x ⊆ z) |
8 | 1, 7 | sylbi 187 | . . . . . 6 ⊢ (z ∈ (M +c N) → ∃x ∈ M x ⊆ z) |
9 | 8 | ancli 534 | . . . . 5 ⊢ (z ∈ (M +c N) → (z ∈ (M +c N) ∧ ∃x ∈ M x ⊆ z)) |
10 | 9 | eximi 1576 | . . . 4 ⊢ (∃z z ∈ (M +c N) → ∃z(z ∈ (M +c N) ∧ ∃x ∈ M x ⊆ z)) |
11 | n0 3559 | . . . 4 ⊢ ((M +c N) ≠ ∅ ↔ ∃z z ∈ (M +c N)) | |
12 | rexcom 2772 | . . . . 5 ⊢ (∃x ∈ M ∃z ∈ (M +c N)x ⊆ z ↔ ∃z ∈ (M +c N)∃x ∈ M x ⊆ z) | |
13 | df-rex 2620 | . . . . 5 ⊢ (∃z ∈ (M +c N)∃x ∈ M x ⊆ z ↔ ∃z(z ∈ (M +c N) ∧ ∃x ∈ M x ⊆ z)) | |
14 | 12, 13 | bitri 240 | . . . 4 ⊢ (∃x ∈ M ∃z ∈ (M +c N)x ⊆ z ↔ ∃z(z ∈ (M +c N) ∧ ∃x ∈ M x ⊆ z)) |
15 | 10, 11, 14 | 3imtr4i 257 | . . 3 ⊢ ((M +c N) ≠ ∅ → ∃x ∈ M ∃z ∈ (M +c N)x ⊆ z) |
16 | 15 | 3ad2ant3 978 | . 2 ⊢ ((M ∈ V ∧ N ∈ W ∧ (M +c N) ≠ ∅) → ∃x ∈ M ∃z ∈ (M +c N)x ⊆ z) |
17 | addcexg 4393 | . . . 4 ⊢ ((M ∈ V ∧ N ∈ W) → (M +c N) ∈ V) | |
18 | brlecg 6112 | . . . 4 ⊢ ((M ∈ V ∧ (M +c N) ∈ V) → (M ≤c (M +c N) ↔ ∃x ∈ M ∃z ∈ (M +c N)x ⊆ z)) | |
19 | 17, 18 | syldan 456 | . . 3 ⊢ ((M ∈ V ∧ N ∈ W) → (M ≤c (M +c N) ↔ ∃x ∈ M ∃z ∈ (M +c N)x ⊆ z)) |
20 | 19 | 3adant3 975 | . 2 ⊢ ((M ∈ V ∧ N ∈ W ∧ (M +c N) ≠ ∅) → (M ≤c (M +c N) ↔ ∃x ∈ M ∃z ∈ (M +c N)x ⊆ z)) |
21 | 16, 20 | mpbird 223 | 1 ⊢ ((M ∈ V ∧ N ∈ W ∧ (M +c N) ≠ ∅) → M ≤c (M +c N)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 Vcvv 2859 ∪ cun 3207 ∩ cin 3208 ⊆ wss 3257 ∅c0 3550 +c cplc 4375 class class class wbr 4639 ≤c clec 6089 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-lec 6099 |
This theorem is referenced by: addlecncs 6209 |
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